nmultichoosek

nmultichoosek is a combinatorial function that counts the number of ways to choose k elements from an n-element set when repetition is allowed and the order of selection is irrelevant. Equivalently, it counts the number of multisets of size k drawn from an n-element universe.

For nonnegative integers n and k, nmultichoosek(n, k) equals the multiset coefficient binomial(n + k - 1, k) = (n + k - 1)! / (k! (n - 1)!). The value for k = 0 is 1 (the empty multiset), while if n = 0 and k > 0 the value is 0. This function is sometimes described as the number of combinations with repetition and is closely related to the stars and bars technique.

In mathematical practice, nmultichoosek can be interpreted as the coefficient of x^k in the expansion of (1 + x + x^2 + ...)^n, reflecting the ways to distribute k indistinguishable items into n distinct bins. It is also referred to as the multiset coefficient (n choose k with repetition) and is denoted in various texts by different notations; some libraries implement it as NMultichooseK or nmultichoosek with the parameters in the order (n, k).

Examples: nmultichoosek(5, 3) = binomial(7, 3) = 35, and nmultichoosek(4, 2) = binomial(5, 2) = 10. Applications include counting multisets in probability, combinatorial enumeration, and problems involving selection with repetition.