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.

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 $\sum \binom{n}{k_1, \dots, k_m} = m^n$.

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.