Multinomial Probability Mass Function

Let $\mathbb{N}$ be the set of natural numbers, $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, let $l\in\mathbb{N}$, and let $\mathbb{R}^l$ be Euclidean space, whose points are written $k=(k_1,\dots,k_l)$. Let $\bar S=(\bar S_1,\dots,\bar S_l)$ be a point of the probability simplex $\Delta^l$, whose coordinates we write with subscripts, and let $n\in\mathbb{N}$. Put

$$\mathsf{K}_n=\Bigl\{k\in\mathbb{N}_0^{\,l}:\ \sum_{\gamma=1}^{l}k_\gamma=n\Bigr\}\subseteq\mathbb{R}^l.$$

The multinomial probability mass function with $n$ trials and cell probabilities $\bar S$ is the function $p_{n,\bar S}:\mathbb{R}^l\to\mathbb{R}$,

$$p_{n,\bar S}(k)=\frac{n!}{k_1!\cdots k_l!}\prod_{\gamma=1}^{l}\bar S_\gamma^{\,k_\gamma}\quad(k\in\mathsf{K}_n),\qquad p_{n,\bar S}(k)=0\quad(k\in\mathbb{R}^l\setminus\mathsf{K}_n),$$

with the factorial (convention $0!=1$), the convention $x^{0}=1$, and the finite product notation.