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Multinomial coefficient

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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:


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