nchoosek

nchoosek is the binomial coefficient C(n, k) in combinatorics, and is also the name of a common function in programming environments such as MATLAB. It counts the number of ways to choose k elements from a set of n distinct elements, ignoring order. For nonnegative integers, C(n, k) = n! / (k! (n−k)!). It is zero if k < 0 or k > n, and it satisfies the symmetry C(n, k) = C(n, n−k) and Pascal’s rule C(n, k) = C(n−1, k−1) + C(n−1, k). The binomial theorem expresses (1 + x)^n as the sum from k = 0 to n of C(n, k) x^k.

Extensions and generalizations: For non-integer n or k, the generalized binomial coefficient is defined via the gamma function: C(n, k) = Γ(n+1) / (Γ(k+1) Γ(n−k+1)). This extends the concept beyond integers, though the combinatorial interpretation changes accordingly.

Computational considerations: C(n, k) grows rapidly with n and k and can overflow for large values. Numeric methods include using logarithms, recurrence relations, or gamma-based evaluation to maintain stability in calculations.

Applications: The binomial coefficient appears in counting k-element subsets of an n-element set, in probability (binomial distributions), and in various areas of algebra and combinatorics. It also forms the coefficients of expanded polynomials and plays a key role in combinatorial identities and algorithms.

nchoosek in MATLAB: The function nchoosek(n, k) returns the binomial coefficient for scalar inputs. If n is a vector or matrix, and k is a scalar (or compatible), it returns all k-element combinations from the elements of n, enabling generation of combination sets. See also factorial, gamma function, binomial distribution, and Pascal’s triangle.