n
Σ C(n,k) x(n-k) yk
k=0
where x and y are variables and n is a nonnegative integer.
Ex. (x + y)4
1x4y0 + 4x3y1 + 6x2y2 + 4x1y3 + 1x0y4
Corollary 1
n
Σ C(n,k) = 2n
k=0
Pascal's Identity
C(n+1,k) = C(n,k-1) + C(n,k)
where n and k are positive integers with n >= k.
Pascal's Triangle
n=0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1
7 1 7 21 35 35 21 7 1
8 1 8 28 56 70 56 28 8 1
9 1 9 36 84 126 126 84 36 9 1
10 1 10 45 120 210 252 210 120 45 10 1
11 1 11 55 165 330 462 462 330 165 55 11 1
12 1 12 66 220 495 792 924 792 495 220 66 12 1