nchoosekn

nchoosekn is a computational function commonly used to compute the binomial coefficient, denoted C(n, k) or nCk. It represents the number of ways to choose k items from a set of n distinct items, disregarding order. In mathematical terms, C(n, k) = n! / (k! (n − k)!). The function typically returns 0 when k < 0 or k > n, and returns 1 when k equals 0 or n. Because of symmetry, C(n, k) = C(n, n − k), which is often exploited to reduce the number of operations in implementations.

Various algorithms exist to implement nchoosekn. A direct factorial approach computes factorials, but it can be inefficient for large n due to rapidly growing intermediate values. A more robust method uses a multiplicative formula: result = product from i = 1 to m of (n − m + i) / i, where m = min(k, n − k). This approach minimizes intermediate overflow by performing reductions and using integer arithmetic or arbitrary-precision integers. In modular contexts, results may be computed modulo a prime using precomputed factorials and modular inverses. Dynamic programming approaches, such as Pascal’s triangle, provide alternative implementations with different space-time trade-offs.

nchoosekn appears in numerous domains, including combinatorial counting, probability calculations, statistics, and algorithms that rely on binomial expansions. It is closely related to concepts like the binomial coefficient and Pascal’s triangle.