Combinations Meet Pascal's Triangle

The combination formula C(n,r) builds Pascal's triangle — the numerical pattern where each row begins and ends with 1, and each interior entry is the sum of the two above it. Row 5 reads 1, 5, 10, 10, 5, 1 — which is exactly C(5,0) through C(5,5). Pascal's triangle is combinations visualized.

The triangle also encodes the binomial expansion: (a + b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵. Each coefficient comes from row 5. Every time a statistician asks "what's the chance of exactly 3 heads in 5 flips?" they reach for C(5,3) = 10, divided by 2⁵ = 32, giving 10/32 ≈ 31%. One triangle ties probability, algebra, and counting together.