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Difference between revisions of "Multinomial coefficient"

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The coefficient
 
The coefficient
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m0653201.png" /></td> </tr></table>
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$$\frac{n!}{n!\dots n_m!},\quad n_1+\ldots+n_m=n,$$
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m0653202.png" /> in the expansion of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m0653203.png" />. In combinatorics, the multinomial coefficient expresses the following: a) the number of possible permutations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m0653204.png" /> elements of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m0653205.png" /> are of one form, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m0653206.png" /> of another form<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m0653207.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m0653208.png" />-th form; b) the number of ways of locating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m0653209.png" /> different elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m06532010.png" /> different cells in which cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m06532011.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m06532012.png" /> elements, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065320/m06532013.png" />, without taking the order of the elements in any cell into account.
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of $x_1^{n_1}\dots x_m^{n_m}$ in the expansion of the polynomial $(x_1+\ldots+x_m)^n$. 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$,\dots,n_m$ 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, $i=1,\dots,m$, without taking the order of the elements in any cell into account.
  
 
Particular cases of multinomial coefficients are the [[Binomial coefficients|binomial coefficients]].
 
Particular cases of multinomial coefficients are the [[Binomial coefficients|binomial coefficients]].

Revision as of 13:53, 15 September 2014

The coefficient

$$\frac{n!}{n!\dots n_m!},\quad n_1+\ldots+n_m=n,$$

of $x_1^{n_1}\dots x_m^{n_m}$ in the expansion of the polynomial $(x_1+\ldots+x_m)^n$. 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$,\dots,n_m$ 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, $i=1,\dots,m$, without taking the order of the elements in any cell into account.

Particular cases of multinomial coefficients are the binomial coefficients.

References

[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: http://encyclopediaofmath.org/index.php?title=Multinomial_coefficient&oldid=18170
This article was adapted from an original article by S.A. Rukova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article