How the triangle is built
Rows and positions are numbered from 0. Each row starts and ends with 1, and every inner entry is the sum of the two entries above it: C(n, k) = C(n − 1, k − 1) + C(n − 1, k).
The entry in row n, position k is the binomial coefficient C(n, k) = n! ÷ (k! × (n − k)!). It is the number of ways to choose k items from n, and the coefficient of xᵏyⁿ⁻ᵏ in (x + y)ⁿ.
Worked example
Row 5 is 1, 5, 10, 10, 5, 1, so (x + y)⁵ = x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵. Its entries add to 2⁵ = 32.
C(52, 5) = 2,598,960 is the number of different five-card hands from a 52-card deck.
Useful patterns
Each row reads the same forwards and backwards, because C(n, k) = C(n, n − k). Row n adds up to 2ⁿ. The second diagonal holds the counting numbers, the third holds the triangular numbers, and adding along the shallow diagonals gives the Fibonacci numbers.