Binomial Coefficients & Pascal's Triangle

Write 1 at the top. Below it, two 1s. Keep going, each new number being the sum of the two just above it: 1, then 1-2-1, then 1-3-3-1. This is Pascal's Triangle, and it is secretly a table of combinations — the entry in row n, position k is exactly C(n, k), the number of ways to choose k things from n.