What is Pascal’s Triangle?
Pascal’s Triangle is a triangular arrangement of binomial coefficients. Each number is the sum of the two numbers directly above it (treating missing values as 0). The nth row corresponds to the coefficients in the expansion of (a + b)^n.
Pascal’s Triangle table
The table below shows the first 16 rows (row 0 through row 15). Use it for quick coefficient lookup.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 11 55 165 330 462 462 330 165 55 11 1
1 12 66 220 495 792 924 792 495 220 66 12 1
1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1
1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1
How to read the triangle
- Rows are indexed from 0. Row
nhasn + 1values. - The first and last value in every row is
1. - Interior values follow:
C(n, k) = C(n-1, k-1) + C(n-1, k).
Common patterns
- Row sums: the sum of values in row
nequals2^n. - Diagonal ones: edges are always
1. - Counting numbers: the second diagonal is
1, 2, 3, 4, ... - Triangular numbers: the third diagonal is
1, 3, 6, 10, 15, ...
Binomial expansion
The coefficients of (a + b)^n are the values in row n:
(a + b)^n = Σ C(n, k) a^(n-k) b^k, k = 0..n