Multinomial Theorem

Let $\mathbb{N}$ be the set of natural numbers, write $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ for the nonnegative integers, and let $\mathbb{R}$ be the set of real numbers. We use the factorial $k!$ for $k\in\mathbb{N}$ together with the conventions $0!=1$ and $x^{0}=1$ for every real $x$, and the finite product notation.

Let $m\in\mathbb{N}$, let $a_1,\dots,a_m\in\mathbb{R}$, and let $d\in\mathbb{N}_0$. Then

$$(a_1+\dots+a_m)^{d}=\sum_{\substack{(n_1,\dots,n_m)\in\mathbb{N}_0^{m}\\ n_1+\dots+n_m=d}}\frac{d!}{n_1!\cdots n_m!}\,\prod_{j=1}^{m}a_j^{\,n_j}.$$

The sum is over the finitely many $m$-tuples of nonnegative integers with entrywise $n_j\le d$ summing to $d$.