## You are here

Homemultinomial theorem

## Primary tabs

# 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

$\binom{n}{n_{1},n_{2},\ldots,n_{k}}.$ |

Defines:

multinomial, multinomial coefficient

Keywords:

multinomial

Related:

BinomialFormula, BinomialCoefficient, GeneralizedLeibnizRule, NthDerivativeOfADeterminant

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

05A10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections