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Math 2.0

Combinatorics Basic counting Theorem

Multinomial theorem

Builds on Multinomial coefficient

Statement

Let $R$ be a commutative ring, let $x_1, \dots, x_m \in R$ , and let $n$ be a natural number. Then $(x_1 + \dots + x_m)^n = \sum_{k_1 + \dots + k_m = n} \binom{n}{k_1, \dots, k_m} x_1^{k_1} \cdots x_m^{k_m}.$ The sum runs over all $m$ -tuples $(k_1, \dots, k_m)$ of natural numbers with sum $n$ , and the coefficients are multinomial coefficients.

Remarks

For $m = 2$ it is the binomial theorem. Example: $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2xz$ . With all $x_i = 1$ it gives $\sum \binom{n}{k_1, \dots, k_m} = m^n$ .

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Multinomial coefficient definition
Multinomial theorem statement

Further reading

Web https://en.wikipedia.org/wiki/Multinomial_theorem, Theorem

About this page

Tag	`0XFC1VQYEK`
URI	`m2://0XFC1VQYEK`
Kind	statement, theorem
Writing	W1 stated
MSC	05A10
Status	This copy of the site has no build data. Run `m2 status 0XFC1VQYEK`.
Uses	Multinomial coefficient
Same as	Wikidata Q619985 Wikipedia Multinomial theorem

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