Block Designs and the Fano Plane (2-(7,3) and Its Complement)

A 2-(v,k,λ) design is a family of k-subsets (blocks) of a v-set in which every pair of points lies in exactly λ blocks. Two counting identities fix r and b; the Fano plane is the 2-(7,3,1) design, and complementing its blocks gives the 2-(7,4,2) design.