Multinomial Coefficient. In particular the coefficient of x 1 b 1 x 2 b 2 x k b k x 1 b 1 x 2 b 2 cdots x k b k x 1 b 1 x 2 b 2 x k b k is n b 1 b 2 b k binom n b 1 b 2 ldots b k b 1 b 2 b k n. The multinomial coefficients the multinomial coefficient is widely used in statistics for example when computing probabilities with the hypergeometric distribution.
In formal terms the multinomial coefficient formula gives the expansion of k 1 k 2 k n where k i are non negative integers. This multinomial coefficient gives the number of ways of depositing 4 distinct objects into 3 distinct groups with i objects in the first group j objects in the second group and k objects in the third group when the order in which they are deposited doesn t matter. This is best illustrated with an example.
The expected number of times the outcome i was observed over n trials is.
The multinomial coefficient is also the number of distinct ways to permute a multiset of n elements and ki are the multiplicities of each of the distinct elements. Multinomial coefficient n. A multinomial coefficient describes the number of possible partitions of n objects into k groups of size n1 n2 nk. The multinomial coefficients are the coefficients of the terms in the expansion of x 1 x 2 x k n x 1 x 2 cdots x k n x 1 x 2 x k n.
