ncm

nCm, read "n choose m," is the binomial coefficient. It counts the number of ways to select m elements from a set of n distinct elements without regard to order. For integers n ≥ 0 and 0 ≤ m ≤ n, it is defined as nCm = n!/(m!(n−m)!). If m is outside the range 0 ≤ m ≤ n, the conventional value is 0.

The notation is commonly written as C(n, m) or nCm. The binomial coefficient is symmetric: C(n, m) = C(n, n−m), since choosing m elements is equivalent to omitting n−m elements.

It also satisfies a simple recurrence: C(n, m) = C(n−1, m) + C(n−1, m−1), with base cases C(n, 0) = C(n, n) = 1. These relations underpin Pascal's triangle, which tabulates C(n, m) for 0 ≤ m ≤ n.

In the binomial theorem, (x + y)^n = sum from m = 0 to n of C(n, m) x^(n−m) y^m, linking binomial coefficients to polynomial expansions. Combinatorially, C(n, m) counts the number of m-element subsets of an n-element set, providing a direct interpretation.

Generalizations extend to non-integer n via the Gamma function: nCm = Γ(n+1)/(Γ(m+1) Γ(n−m+1)). In probability and statistics, binomial coefficients appear in distributions such as the binomial and hypergeometric distributions, and in counting problems across mathematics.