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# Multinomial coefficient  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

In discrete mathematics, the multinomial coefficient arises as a generalization of the binomial coefficient.

Let k1, k2, ..., km be natural numbers giving a partition of n: The multinomial coefficient is defined by For m = 2 we may write: so that It follows that the multinomial coefficient is equal to the binomial coefficient for the partition of n into two integer numbers. However, the two coefficients (binomial and multinomial) are notated somewhat differently for m = 2.

The multinomial coefficients arise in the multinomial expansion The number of terms in this expansion is equal to the binomial coefficient: Example.   Expand (x + y + z)4: The 15 terms are the following: A multinomial coefficient can be expressed in terms of binomial coefficients: ## Reference

D. E. Knuth, The Art of Computer Programming, Vol I. Addison-Wesley, Reading Mass (1968) p. 64