# multinomial theorem

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

$${a}_{1}{x}_{1}+{a}_{2}{x}_{2}+\mathrm{\dots}+{a}_{k}{x}_{k}.$$ |

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:

$${({x}_{1}+{x}_{2}+\mathrm{\dots}+{x}_{k})}^{n}=\sum \frac{n!}{{n}_{1}!{n}_{2}!\mathrm{\cdots}{n}_{k}!}{x}_{1}^{{n}_{1}}{x}_{2}^{{n}_{2}}\mathrm{\cdots}{x}_{k}^{{n}_{k}}$$ | (1) |

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

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

$$\left(\genfrac{}{}{0pt}{}{n}{{n}_{1},{n}_{2},\mathrm{\dots},{n}_{k}}\right).$$ |

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 |