Multivariate hypergeometric distribution ($\mathbb{P}(K_1=k_1,\dots ,K_m=k_m)=\frac{{\prod }_i\left(\genfrac{}{}{0ex}{}{c_i}{k_i}\right)}{\left(\genfrac{}{}{0ex}{}{n}{r}\right)}$)

Suppose a finite population is split into $m$ categories with fixed sizes $c_1,\dots ,c_m$ summing to $n$. A uniform size-$r$ sample drawn by sampling without replacement has category counts $K_1,\dots ,K_m$ with the displayed joint probability mass function, for integers $0\le k_i\le c_i$ 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 probabilityPut $p_i=c_i/n$. When $n>1$, its covariance matrix is the corresponding multinomial distribution covariance matrix multiplied by the squared finite population correction: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.^[ref]