 Combinatorial design theory

Combinatorial design theory studies finite incidence patterns in which prescribed small collections occur equally often. A $t$-$(v,k,\lambda )$ combinatorial design consists of $v$ points and blocks of size $k$, with every $t$-element subset of points in exactly $\lambda$ blocks. Counting incidences between blocks and their $t$-element subsets giveswhere $b$ is the number of blocks.^[ref]

On $\{1,2,3,4\}$, take all four three-element subsets as blocks. Each pair occurs twice, so this is a $2$-$(4,3,2)$ combinatorial design; the count is $4\cdot 3=2\cdot 6$. These exact balance conditions turn questions about fair comparisons and repeated pairings into existence, construction and counting problems. The displayed divisibility condition is necessary, but does not by itself ensure that a design exists.