Multinomial coefficient

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The coefficient

$$\frac{n!}{n_1!\dotsm n_m!},\quad n_1+\dotsb+n_m=n,$$

of $x_1^{n_1}\dotsm x_m^{n_m}$ in the expansion of the polynomial $(x_1+\dotsb+x_m)^n$. In combinatorics, the multinomial coefficient expresses the following: a) the number of possible permutations of $n$ elements of which $n_1$ are of one form, $n_2$ of another form$,\dotsc,n_m$ of the $m$-th form; b) the number of ways of locating $n$ different elements in $m$ different cells in which cell $i$ contains $n_i$ elements, $i=1,\dotsc,m$, without taking the order of the elements in any cell into account.

Particular cases of multinomial coefficients are the binomial coefficients.


[1] M. Hall, "Combinatorial theory" , Wiley (1986)
[2] J. Riordan, "An introduction to combinatorial analysis" , Wiley (1967)
How to Cite This Entry:
Multinomial coefficient. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.A. Rukova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article