Tổng của hàng thứ 3 trong tam giác pascal là bao nhiêu?

Tam giác Pascal là một mảng số đại diện cho một mẫu số. Nó có một số cách sử dụng khác nhau trong toán học và thống kê, nhưng trong ngữ cảnh của đa thức, cụ thể là nhị thức, nó được sử dụng để mở rộng các nhị thức

Tính chất của tam giác Pascal

Dưới đây là một phần của tam giác Pascal;

  • Tam giác Pascal đối xứng;
  • Các đường chéo ngoài cùng của tam giác Pascal đều là "1. "
  • Các giá trị bên trong tam giác (không phải là 1) được xác định bằng tổng của hai giá trị ngay phía trên và liền kề. Tham khảo hình bên dưới để rõ
    Trong hình trên, 3 ví dụ về mối quan hệ của các giá trị trong tam giác Pascal được hiển thị. Từ trên xuống dưới, màu vàng, hai giá trị là 1 và 1, tổng bằng 2, giá trị bên dưới. Tương tự, 3 + 1 = 4 màu cam và 4 + 6 = 10 màu xanh lam
  • Tổng các số trong mỗi hàng của tam giác Pascal bằng 2n trong đó n đại diện cho số hàng trong tam giác Pascal bắt đầu từ n=0 cho hàng đầu tiên ở trên cùng. Tham khảo hình bên dưới để rõ

Có những tính chất khác của tam giác Pascal ngoài những tính chất được liệt kê ở trên, nhưng việc hiểu những tính chất được liệt kê ở trên có thể hữu ích khi sử dụng tam giác Pascal để mở rộng các nhị thức

khai triển nhị thức

Có thể sử dụng tam giác Pascal để xác định các hệ số khi khai triển nhị thức. Cụ thể, hệ số nhị thức, thường được viết là , cho chúng ta biết mục thứ b của hàng thứ n của tam giác Pascal; .

Ví dụ

Mở rộng. (x + y)3

Tham khảo hình sau cùng với lời giải thích bên dưới

Trong ví dụ này, n = 3, biểu thị hàng thứ 4 của tam giác Pascal (vì hàng đầu tiên là n = 0). Các số trong hàng, 1 3 3 1, là các hệ số và b cho biết chúng ta đang đề cập đến hệ số nào trong hàng. Tham khảo lại ví dụ trên. Hệ số của số hạng đầu tiên, x3, là ở b = 0 của hàng n = 3, hoặc 1. Tương tự cho mỗi số hạng tương ứng sao cho hệ số của các số hạng thứ 2, 3 và 4 lần lượt là 3, 3 và 1, đúng như ở hàng n = 3 của tam giác Pascal

Tiếp theo, chúng ta có thể xác định giá trị của các biểu thức nhân với từng hệ số. Mỗi số hạng có một số thành phần của x và một số thành phần của y được nâng lên thành số mũ. Số mũ trên các thành phần x và y có tổng bằng n. Bắt đầu từ bên trái, x có số mũ bằng n hoặc 3 và y có số mũ bằng 0. Di chuyển từ trái sang phải, 1 bị trừ khỏi số mũ trên thành phần x trong khi 1 được thêm vào số mũ trên thành phần y, dẫn đến số hạng cuối cùng có số mũ là 0 trên thành phần x và số mũ là 3 trên thành phần x. . Điều này có thể được nhìn thấy trong ví dụ trên, trong đó các số mũ trên mỗi thuật ngữ được viết rõ ràng

Sử dụng tam giác Pascal và các mẫu này, chúng ta có thể mở rộng các nhị thức được nâng lên lũy thừa bậc n mà nếu không sẽ rất tẻ nhạt để mở rộng thông qua phép nhân lặp đi lặp lại. Tham khảo trang định lý nhị thức để biết cách tiếp cận công thức để khai triển các nhị thức, cách này thậm chí còn hiệu quả hơn khi bạn đã quen thuộc với tất cả các ký hiệu toán học trong công thức

Cách tiếp cận ngây thơ. Trong tam giác Pascal, mỗi mục của một hàng là giá trị của hệ số nhị thức. Vì vậy, một giải pháp đơn giản là tạo tất cả các phần tử hàng cho đến hàng thứ n và thêm chúng. Nhưng cách tiếp cận này sẽ có độ phức tạp thời gian O(n3). Tuy nhiên, nó có thể được tối ưu hóa lên đến độ phức tạp thời gian O(n2). Tham khảo bài viết sau để tạo các phần tử của tam giác Pascal.  

  • tam giác Pascal

Giải pháp tốt hơn. Cùng tìm hiểu về mẫu tam giác của pascal

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210

Như hình trên tổng các phần tử ở hàng thứ i bằng 2i. Giờ đây, có thể dễ dàng tính tổng của tất cả các phần tử cho đến hàng thứ n bằng cách cộng các lũy thừa của 2

Dưới đây là việc thực hiện phương pháp trên.  

C++




// C++ program to find sum of all elements

// upto nth row in Pascal triangle.

#include

using namespace

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
0

 

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
1

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
2

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
3
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
3
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
5
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
6
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
5
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
8

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
9

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
1

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
3
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
3
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
5
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
6

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
8

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
0

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
2
Sum of all elements:1023
3
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
5
Sum of all elements:1023
5

Sum of all elements:1023
6_______21_______7

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
9

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
1
Sum of all elements:1023
2

Sum of all elements:1023
9

 

Sum of all elements:1023
4

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
5
Sum of all elements:1023
6

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
9

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
5 // C++ program to find sum of all elements0

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0// C++ program to find sum of all elements2// C++ program to find sum of all elements3 // C++ program to find sum of all elements4

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
1 // C++ program to find sum of all elements7

Sum of all elements:1023
9

Java




// C++ program to find sum of all elements9

// upto nth row in Pascal triangle.

// upto nth row in Pascal triangle.1 // upto nth row in Pascal triangle.2

 

// upto nth row in Pascal triangle.3 // upto nth row in Pascal triangle.4

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
1

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
2

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0#include 0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
3
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
6
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
5
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
8

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
9

 

Sum of all elements:1023
6
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
1

Sum of all elements:1023
6_______1_______3 using1using2using3

 

Sum of all elements:1023
6_______0_______8

Sum of all elements:1023
6_______21_______0

Sum of all elements:1023
6_______21_______2
Sum of all elements:1023
3
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
5 namespace2using2namespace4

namespace5_______288_______6namespace7 namespace8

Sum of all elements:1023
6_______21_______9

 

Sum of all elements:1023
6_______21_______1
Sum of all elements:1023
2

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
9

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
07

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
09 #include 0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
11
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
12

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
9

Sum of all elements:1023
6_______1_______5
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
17_______1_______18using3

Sum of all elements:1023
6_______1_______21
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
22

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
23
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
24

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
9

Sum of all elements:1023
9

Python3




                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
28

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
29

 

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
30

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
31

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
32
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
33

Sum of all elements:1023
6

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
36

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
38
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
39 using2

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
43

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
45

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
2
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
48
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
49
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
50
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
51

Sum of all elements:1023
6_______1_______38
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
39
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
38
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
56
Sum of all elements:1023
3namespace7
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
59

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
1
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
38

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
64

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
65
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
39
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
18

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
68
Sum of all elements:1023
3
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
22
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
71

C#




                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
72

// upto nth row in Pascal triangle.

using

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
75

 

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
09 // upto nth row in Pascal triangle.3 // upto nth row in Pascal triangle.4

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
1

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
2

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0#include 0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
3
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
6
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
5
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
8

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
9

 

Sum of all elements:1023
6
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
1

Sum of all elements:1023
6_______1_______3
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
6

 

Sum of all elements:1023
6_______0_______8

Sum of all elements:1023
6_______21_______0

Sum of all elements:1023
6_______21_______2
Sum of all elements:1023
3
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
5
Sum of all elements:1023
5

namespace5

Sum of all elements:1023
7

Sum of all elements:1023
6_______21_______9

 

Sum of all elements:1023
6_______21_______1
Sum of all elements:1023
2

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
9

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0#include 0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
09
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
11
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
18

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
9

Sum of all elements:1023
6_______1_______5 // C++ program to find sum of all elements0

Sum of all elements:1023
6_______0_______25
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
22

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
27______1_______24

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
9

Sum of all elements:1023
9

PHP




                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
32

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
33

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
34

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
35

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
36

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
37

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
38
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
6
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
40
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
41

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
9

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
1

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
46
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
47

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
49

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
51

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
2
Sum of all elements:1023
3_______0_______55
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
47
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
55
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
58
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
40using3
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
55
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
62

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
9

Sum of all elements:1023
6_______0_______46
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
39_______0_______46
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
69
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
55
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
71

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
9

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
1
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
46using3

Sum of all elements:1023
9

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
79

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
40
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
81

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
82
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
83
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
84

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
85
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
6
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
40
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
71

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
89

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
90

Javascript




                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
91

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
92

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
34

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
35

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
36

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
37

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
38
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
98

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
9

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
1

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
03

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
49

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
51

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
2
Sum of all elements:1023
10

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
9

Sum of all elements:1023
6_______21_______7

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
9

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
0
Sum of all elements:1023
1
Sum of all elements:1023
2

Sum of all elements:1023
9

 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
79

Sum of all elements:1023
22

Sum of all elements:1023
23
                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210
83
                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            
24

 

Sum of all elements:1023
26

Sum of all elements:1023
27

đầu ra.

Sum of all elements:1023

 

Thời gian phức tạp. TRÊN)
Không gian phụ trợ. O(1) vì nó đang sử dụng không gian cố định cho các biến

giải pháp hiệu quả.  

2n có thể được biểu thị dưới dạng
2n = ( 20 + 21 + 22 + 23 +. . . + 2(n-1) ) + 1
Ví dụ.  
26 = ( 20 + 21 + 22 + 23 + 24 + 25 ) + 1
64 = ( 1 + 2 + 4 + 8 +16 + 32 ) + 1
64 = 63 + 1

Vì vậy, hãy tính 2n thay vì tính mọi lũy thừa của 2 cho đến (n ​​– 1) và từ ví dụ trên, tổng lũy ​​thừa của 2 cho đến (n ​​– 1) sẽ là (2n – 1)

Tổng các hàng trong tam giác Pascal là bao nhiêu?

Tổng mỗi hàng của tam giác Pascal là lũy thừa 2 . Trên thực tế, tổng các mục ở hàng thứ n là 2n.

Tổng của tam giác Pascal là gì?

Tổng các số trên một đường chéo của tam giác Pascal bằng số bên dưới tổng cuối cùng . Ví dụ: 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 3 = 4, 1 + 3 + 6 = 10, v.v.

Tổng của hàng thứ 5 trong Tam giác Pascal là bao nhiêu?

Tổng = 1+4+6+4+1 = 16 .