Combinatorial design ($t\text{-}(v,k,\lambda )$)

A $t$-$(v,k,\lambda )$ combinatorial design is an incidence structure on $v$ points with blocks containing $k$ points, such that every $t$-element subset lies in exactly $\lambda$ blocks. Repeated blocks may be allowed; a simple design has no repetitions. With $b$ blocks, counting pairs consisting of a block and one of its $t$-subsets givesFor a two-design, each point lies in $r=\lambda (v-1)/(k-1)$ blocks.^[ref]

The seven triples $123,145,167,246,257,347,356$ on $\{1,\dots ,7\}$ form a $2$-$(7,3,1)$ design: every pair occurs once, and every point occurs three times. This is the Fano plane. The equation checks $7\cdot 3=1\cdot 21$. Such exact incidence balance makes combinatorial designs useful for arranging comparisons and constructing highly symmetric finite structures.