Multinomial coefficient

A multinomial coefficient counts ways to divide n distinct objects into specified groups of sizes k₁, …, kᵣ, where the sizes sum to n. It equals n!/(k₁!⋯kᵣ!).

Factorial: The product of all positive integers up to a given nonnegative integer, with 0! defined as 1. The numerator and every denominator term are factorials.
Multinomial distribution: A probability distribution for counts of outcomes across several categories in repeated independent trials. Its probability mass function uses multinomial coefficients to count outcome sequences.
Set partition: A division of a set into nonempty, pairwise disjoint subsets whose union is the whole set. A multinomial coefficient counts partitions into labeled groups of prescribed sizes.
Generating function: A formal power series whose coefficients encode a sequence or family of combinatorial counts. Multinomial coefficients occur as coefficients in powers of sums and related series.
Binomial coefficient: The number of ways to choose a fixed-size subset from a finite set. Repeated binomial choices decompose a multinomial count into successive selections.
Multinomial test: A statistical test comparing observed category counts with specified probabilities. The likelihood of observed counts includes their multinomial coefficient.
Ordered selection: A choice of items in which the order of selection matters. Choosing group members one group at a time yields the factorial ratio.
Pascal's identity: The recurrence expressing a binomial coefficient as the sum of two neighboring coefficients. Its multinomial analogue splits counts according to the group receiving a distinguished object.
Stars and bars: A counting method for distributing identical objects among distinguishable groups. It counts distributions with variable group sizes, unlike fixed-size multinomial arrangements.
Anagrams: Words or phrases formed by rearranging the letters of another word or phrase. Repeated letters create groups of indistinguishable objects, giving a multinomial count.

Linked from 7 pages