Solution
= Solution
The <multinomial distribution> has <probability mass function>
$$
\boxed{p(\mathbf y_i\mid\mathbf p_i,n_i)
=\frac{n_i!}{\prod_{k=1}^K y_{ik}!}
\prod_{k=1}^Kp_{ik}^{\,y_{ik}},\qquad
y_{ik}\in\{0,1,\ldots\},\quad\sum_k y_{ik}=n_i.}
$$
It is zero outside this count simplex. The multinomial coefficient counts the individual category sequences giving the same aggregate counts; the probability vector satisfies $\sum_kp_{ik}=1$.