= Combinatorial design
{title2=$t\text{-}(v,k,\lambda)$}
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 gives
$$
b\binom{k}{t}=\lambda\binom{v}{t}.
$$
For a two-design, each point lies in $r=\lambda(v-1)/(k-1)$ blocks.<On t-designs by Dijen K. Ray-Chaudhuri and Richard M. Wilson>[Introduction and preliminaries, printed page 737]{ref}
The seven triples $123,145,167,246,257,347,356$ on $\{1,\ldots,7\}$ 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 $7\cdot3=1\cdot21$. Such exact incidence balance makes <combinatorial designs> useful for arranging comparisons and constructing highly symmetric finite structures.
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