---
tag: 0XFC1VQYEK
kind: statement
role: theorem
name: Multinomial theorem
msc: [05A10]
uses: [00K0RF9BFY]
sources:
  - url: https://en.wikipedia.org/wiki/Multinomial_theorem
    locator: Theorem
same_as:
  wikidata: Q619985
  wikipedia: Multinomial theorem
---

## 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|00K0RF9BFY]].

## Remarks

For $m = 2$ it is the [[binomial theorem|0EHWPWGHB2]].
Example: $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2xz$.
With all $x_i = 1$ it gives $\sum \binom{n}{k_1, \dots, k_m} = m^n$.

## Explanations

### L1 human

To expand a power of a sum of $m$ terms, pick one term from each of the $n$ factors.
The monomial $x_1^{k_1} \cdots x_m^{k_m}$ appears once for each way to decide which factors give which term.