Binomiluku

Binomiluku, or the binomial coefficient, is denoted as binom(n, k) or C(n, k). It counts the number of k-element subsets of an n-element set and thus plays a central role in combinatorics, probability, and algebra.

For nonnegative integers n and k with 0 ≤ k ≤ n, binom(n, k) = n! / (k! (n − k)!). If k < 0 or k > n, binom(n, k) is defined as 0. Key properties include symmetry binom(n, k) = binom(n, n − k) and Pascal’s rule binom(n, k) = binom(n − 1, k) + binom(n − 1, k − 1). Boundary values: binom(n, 0) = binom(n, n) = 1.

An alternative computation is binom(n, k) = n(n − 1) ... (n − k + 1) / k! for integers n ≥ 0 and 0 ≤ k ≤ n.

Generalization: for any real or complex n and nonnegative integer k, binom(n, k) can be defined by the Gamma function: binom(n, k) = Γ(n + 1) / [Γ(k + 1) Γ(n − k + 1)]. This extends the concept beyond integers and underpins the generalized binomial theorem: (x + y)^n = sum_{k=0}^∞ binom(n, k) x^{n−k} y^k, under suitable convergence.

Binomiluku features prominently in Pascal’s triangle, generating functions, and a wide range of counting problems, including distributions of subsets and various probabilistic calculations.