Binom

Binom is a shorthand for the binomial coefficient, typically written as binom(n, k) or C(n, k). It counts the number of ways to choose k elements from an n-element set without regard to order. For nonnegative integers n and k with 0 ≤ k ≤ n, binom(n, k) equals n! / (k!(n−k)!). The coefficient is symmetric: binom(n, k) = binom(n, n−k).

Binom(n, k) also satisfies Pascal's rule: binom(n, k) = binom(n−1, k) + binom(n−1, k−1). This recurrence underpins Pascal's triangle and many combinatorial algorithms.

In probability and statistics, binom(n, k) appears in the binomial distribution. If X follows a binomial distribution with parameters n and p, then the probability of exactly k successes is P(X = k) = binom(n, k) p^k (1−p)^{n−k}. In algebra, binomial coefficients are central to the binomial theorem: (x + y)^n = sum_{k=0}^n binom(n, k) x^{n−k} y^k.

The concept extends beyond integers through the generalized binomial coefficient. For a real or complex α and nonnegative integer k, binom(α, k) = α(α−1)…(α−k+1)/k!, which reduces to the standard form when α is a nonnegative integer.

In computing, binom(n, k) is implemented in many programming environments as a function (for example, as choose(n, k) or math.comb in Python). When n is large, direct factorial calculations can be inefficient or unstable, so multiplicative or logarithmic methods are commonly used.