The Multinomial Coefficient

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $m\in\mathbb{N}$ and $(k_{1},\dots,k_{m})\in\mathbb{N}_{0}^{m}$, and write $k_{1}+\dots+k_{m}=\sum_{i=1}^{m}k_{i}$.

(i) The multinomial coefficient. The multinomial coefficient

$$\binom{k_{1}+\dots+k_{m}}{k_{1},\dots,k_{m}}$$

is the unique $M\in\mathbb{N}_{0}$ with $M\prod_{i=1}^{m}k_{i}!=\bigl(\sum_{i=1}^{m}k_{i}\bigr)!$, given by Existence and Uniqueness of the Multinomial Coefficients (i).