 Inclusion-exclusion for exact block occupancy

Suppose $N$ disjoint blocks each have $m$ elements. The number of $K$-element subsets having exactly $q>0$ elements in at least one block is the displayed sum, with $1\le r\le \min (N,\lfloor K/q\rfloor )$ and invalid binomial coefficients interpreted as zero. For a specified set of $r$ blocks, choose $q$ elements in each and all remaining elements outside those blocks; inclusion-exclusion then counts the union of the exact-occupancy events. Excluding the rest of each specified block is essential.

// Target: algebra.bigb