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 10
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 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 12
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 13
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 13
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 15
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 16
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 15
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 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 19
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 -> 2100
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 -> 2101
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 -> 2100
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 13
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 13
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 15
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 -> 2106
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 -> 2100
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 -> 2108
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 -> 2100
Sum of all elements:10230
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 -> 2100
Sum of all elements:10232
Sum of all elements:10233
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 15
Sum of all elements:10235
Sum of all elements:10236_______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 -> 2100
Sum of all elements:10239
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 -> 2100
Sum of all elements:10231
Sum of all elements:10232
Sum of all elements:10239
Sum of all elements:10234
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 15
Sum of all elements:10236
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 19
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 -> 2100
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 15
// C++ program to find sum of all elements
0sum 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 -> 2100
// C++ program to find sum of all elements
2// C++ program to find sum of all elements
3 // C++ program to find sum of all elements
4sum 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 -> 2100
Sum of all elements:10231
// C++ program to find sum of all elements
7Sum of all elements:10239
Java
// C++ program to find sum of all elements
9
// 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 -> 2100
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 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 -> 2100
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 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 -> 2100
#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 13
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 16
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 15
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 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 -> 2100
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 19
Sum of all elements:10236
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 -> 2101
Sum of all elements:10236_______1_______3
using
1using
2using
3
Sum of all elements:10236_______0_______8
Sum of all elements:10236_______21_______0
Sum of all elements:10236_______21_______2
Sum of all elements:10233
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 15
namespace
2using
2namespace
4namespace
5_______288_______6namespace
7 namespace
8
Sum of all elements:10236_______21_______9
Sum of all elements:10236_______21_______1
Sum of all elements:10232
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 -> 2100
Sum of all elements:10239
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 -> 2100
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 10 107
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 -> 2100
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 10 109
#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 10 111
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 10 112
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 -> 2100
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 19
Sum of all elements:10236_______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 10 117_______1_______18
using
3Sum of all elements:10236_______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 10 122
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 10 123
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 10 124
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 -> 2100
Sum of all elements:10239
Sum of all elements:10239
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 10 128
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 10 129
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 10 130
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 10 131
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 10 132
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 10 133
Sum of all elements:10236
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 -> 2100
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 10 136
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 -> 2100
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 10 138
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 10 139
using
2sum 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 -> 2100
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 -> 2100
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 10 143
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 -> 2100
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 10 145
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 -> 2100
Sum of all elements:10232
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 10 148
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 10 149
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 10 150
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 10 151
Sum of all elements:10236_______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 10 139
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 10 138
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 10 156
Sum of all elements:10233
namespace
7 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 10 159
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 -> 2100
Sum of all elements:10231
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 10 138
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 -> 2100
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 10 164
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 10 165
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 10 139
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 10 118
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 10 168
Sum of all elements:10233
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 10 122
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 10 171
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 10 172
// 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 10 175
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 10 109
// 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 -> 2100
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 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 -> 2100
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 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 -> 2100
#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 13
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 16
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 15
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 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 -> 2100
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 19
Sum of all elements:10236
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 -> 2101
Sum of all elements:10236_______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 -> 2106
Sum of all elements:10236_______0_______8
Sum of all elements:10236_______21_______0
Sum of all elements:10236_______21_______2
Sum of all elements:10233
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 15
Sum of all elements:10235
namespace
5
Sum of all elements:10237
Sum of all elements:10236_______21_______9
Sum of all elements:10236_______21_______1
Sum of all elements:10232
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 -> 2100
Sum of all elements:10239
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 -> 2100
#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 10 109
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 10 111
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 -> 21018
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 -> 2100
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 19
Sum of all elements:10236_______1_______5
// C++ program to find sum of all elements
0Sum of all elements:10236_______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 10 122
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 -> 21027______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 -> 2100
Sum of all elements:10239
Sum of all elements:10239
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 -> 21032
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 -> 21033
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 -> 21034
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 -> 21035
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 -> 21036
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 -> 21037
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 -> 21038
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 16
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 -> 21040
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 -> 21041
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 19
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 -> 2100
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 -> 2101
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 -> 2100
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 -> 21046
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 -> 21047
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 -> 2100
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 -> 21049
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 -> 2100
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 -> 21051
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 -> 2100
Sum of all elements:10232
Sum of all elements:10233_______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 -> 21047
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 -> 21055
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 -> 21058
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 -> 21040
using
3sum 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 -> 21055
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 -> 21062
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 -> 2100
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 19
Sum of all elements:10236_______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 10 139_______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 -> 21069
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 -> 21055
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 -> 21071
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 -> 2100
Sum of all elements:10239
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 -> 2100
Sum of all elements:10231
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 -> 21046
using
3Sum of all elements:10239
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 -> 21079
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 -> 21040
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 -> 21081
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 -> 21082
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 -> 21083
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 -> 21084
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 -> 21085
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 16
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 -> 21040
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 -> 21071
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 -> 21089
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 -> 21090
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 -> 21091
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 -> 21092
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 -> 21034
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 -> 21035
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 -> 21036
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 -> 21037
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 -> 21038
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 -> 21098
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 19
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 -> 2100
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 -> 2101
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 -> 2100
Sum of all elements:102303
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 -> 2100
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 -> 21049
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 -> 2100
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 -> 21051
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 -> 2100
Sum of all elements:10232
Sum of all elements:102310
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 -> 2100
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 19
Sum of all elements:10236_______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 -> 2100
Sum of all elements:10239
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 -> 2100
Sum of all elements:10231
Sum of all elements:10232
Sum of all elements:10239
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 -> 21079
Sum of all elements:102322
Sum of all elements:102323
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 -> 21083
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 10 124
Sum of all elements:102326
Sum of all elements:102327
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]