# multinomial theorem

A multinomial is a mathematical expression consisting of two or more terms, e.g.

 $a_{1}x_{1}+a_{2}x_{2}+\ldots+a_{k}x_{k}.$

The multinomial theorem provides the general form of the expansion of the powers of this expression, in the process specifying the multinomial coefficients  which are found in that expansion. The expansion is:

 $(x_{1}+x_{2}+\ldots+x_{k})^{n}=\sum\frac{n!}{n_{1}!n_{2}!\cdots n_{k}!}x_{1}^{% n_{1}}x_{2}^{n_{2}}\cdots x_{k}^{n_{k}}$ (1)

where the sum is taken over all multi-indices $(n_{1},\ldots n_{k})\in\mathbb{N}^{k}$ that sum to $n$.

The expression $\frac{n!}{n_{1}!n_{2}!\cdots n_{k}!}$ occurring in the expansion is called multinomial coefficient and is denoted by

 $\binom{n}{n_{1},n_{2},\ldots,n_{k}}.$
 Title multinomial theorem Canonical name MultinomialTheorem Date of creation 2013-03-22 13:13:05 Last modified on 2013-03-22 13:13:05 Owner bshanks (153) Last modified by bshanks (153) Numerical id 12 Author bshanks (153) Entry type Theorem Classification msc 05A10 Related topic BinomialFormula Related topic BinomialCoefficient Related topic GeneralizedLeibnizRule Related topic NthDerivativeOfADeterminant Defines multinomial Defines multinomial coefficient