binomnnk

Binomnnk is a notational variant used in some mathematical texts to denote the binomial coefficient, commonly written as C(n, k) and read as “n choose k.” For nonnegative integers n and k with 0 ≤ k ≤ n, binomnnk(n, k) equals n!/(k!(n−k)!). It counts the number of k-element subsets of an n-element set and appears as the coefficient of x^k in the expansion of (1+x)^n.

Key properties include symmetry binomnnk(n, k) = binomnnk(n, n−k), and Pascal’s identity binomnnk(n, k) = binomnnk(n−1, k−1) + binomnnk(n−1, k). The boundary values binomnnk(n, 0) = binomnnk(n, n) = 1 hold for all n ≥ 0. In combinatorics, binomnnk counts subsets; in algebra, it is the coefficient in the binomial theorem.

Generalizations exist beyond integers: for real or complex n and integer k, binomnnk(n, k) can be defined as Γ(n+1)/(Γ(k+1)Γ(n−k+1)). The q-binomial coefficient is a different generalization used in combinatorics and number theory.

Computationally, a common form is the multiplicative formula binomnnk(n, k) = n(n−1)…(n−k+1)/k!, which is stable for moderate sizes when computed with cancellation. The term binomnnk is not standard; many sources simply write binom(n, k) or C(n, k).