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# Multinomial coefficient

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

Let *k*_{1}, *k*_{2}, ..., *k*_{m} 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