Multinomial theorem

In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials.

Theorem

For any positive integer m and any non-negative integer n, the multinomial theorem describes how a sum with m terms expands when raised to the nth power:

(

x

1

+

x

2

+

⋯

+

x

m

)

n

=

∑

k

1

+

k

2

+

⋯

+

k

m

=

n

k

1

,

k

2

,

⋯

,

k

m

≥

0

(

n

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1

,

k

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,

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k

m

)

x

1

k

1

⋅

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2

⋯

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m

{\displaystyle (x_{1}+x_{2}+\cdots +x_{m})^{n}=\sum _{\begin{array}{c}k_{1}+k_{2}+\cdots +k_{m}=n\\k_{1},k_{2},\cdots ,k_{m}\geq 0\end{array}}{n \choose k_{1},k_{2},\ldots ,k_{m}}x_{1}^{k_{1}}\cdot x_{2}^{k_{2}}\cdots x_{m}^{k_{m}}}

where

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k

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{\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}}

is a multinomial coefficient. The sum is taken over all combinations of nonnegative integer indices k1 through km such that the sum of all ki is n. That is, for each term in the expansion, the exponents of the xi must add up to n.

In the case m = 2, this statement reduces to that of the binomial theorem.