= Multivariate hypergeometric distribution
{title2=$\mathbb P(K_1=k_1,\ldots,K_m=k_m)=\frac{\prod_i\binom{c_i}{k_i}}{\binom nr}$}
Suppose a finite population is split into $m$ categories with fixed sizes $c_1,\ldots,c_m$ summing to $n$. A uniform size-$r$ sample drawn by <sampling without replacement> has category counts $K_1,\ldots,K_m$ with the displayed <joint probability mass function>, for <integers> $0\leq k_i\leq c_i$ summing to $r$. It extends the two-category <hypergeometric distribution>; replacing the sampled objects instead gives a <multinomial distribution>.
For a box with category sizes $(5,3,2)$ and a sample of four distinct objects, the count pattern $(2,1,1)$ has <probability>
$$
\frac{\binom52\binom31\binom21}{\binom{10}4}
=\frac{60}{210}=\frac27.
$$
Put $p_i=c_i/n$. When $n>1$, its <covariance matrix> is the corresponding <multinomial distribution> <covariance matrix> multiplied by the squared <finite population correction>:
$$
\operatorname{Var}(K_i)=rp_i(1-p_i)\frac{n-r}{n-1},
\qquad
\operatorname{Cov}(K_i,K_j)=-rp_ip_j\frac{n-r}{n-1}\quad(i\ne j).
$$
This connects the fixed-total dependence between categories with the further reduction from <sampling without replacement>. X. G. Duan gives a symmetry-based explanation of this shared <covariance> structure.https://arxiv.org/abs/2101.00548[Duan, Better understanding of the multivariate hypergeometric distribution, 2021 preprint]{ref}
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