A multinomial is a mathematical expression consisting of two or more terms, e.g. $$a_1 x_1 + a_2 x_2 + \ldots + a_k x_k.$$ The multinomial theorem provides the general form of the expansion of the powers of this expression, in the process specifying the multinomial coefficients which are found in that expansion. The expansion is: \begin{equation} (x_1 + x_2 + \ldots + x_k)^n = \sum \frac{n!}{n_1! n_2! \dotsb n_k!} x_1^{n_1} x_2^{n_2} \cdots x_k^{n_k} \end{equation}where the sum is taken over all multi-indices $(n_1, \ldots n_k)\in\mathbb{N}^k$ that sum to $n$

The expression $\frac{n!}{n_1! n_2! \cdots n_k!}$ occurring in the expansion is called multinomial coefficient and is denoted by \begin{equation*} \binom{n}{n_1, n_2, \dotsc, n_k}. \end{equation*}