determine b and r, and not all combinations of v, k, and λ are possible. The two basic equations connecting these parameters are obtained by counting the number of pairs (B, p) where B is a block and p is a point in that block, and obtained from counting for a fixed x the triples (x, y, B) where x and y are distinct points and B is a block that contains them both. This equation for every x also proves that r is constant (independent of x) even without assuming it explicitly, thus proving that the condition that any x in X is contained in r blocks is redundant and r can be computed from the other parameters. These conditions are not sufficient as, for example, a (43,7,1)-design does not exist. The order of a 2-design is defined to be n = r − λ. The complement of a 2-design is obtained by replacing each block with its complement in the point set X. It is also a 2-design and has parameters v′ = v, b′ = b, r′ = b − r, k′ = v − k, λ′ = λ + b − 2r. A 2-design and its complement have the same order. A fundamental theorem, Fisher's inequality, named after the statistician Ronald Fisher, is that b ≥ v in any 2-design. Examples The unique (6,3,2)-design (v = 6, k = 3, λ = 2) has 10 blocks (b = 10) and each element is repeated 5 times (r = 5). Using the symbols 0 − 5, the blocks are the following triples: 012 013 024 035 045 125 134 145 234 235. and the corresponding incidence matrix (a v×b binary matrix with constant row sum r and constant column sum k) is: One of four nonisomorphic (8,4,3)-designs has 14 blocks with each element repeated 7 times. Using the symbols 0 − 7 the blocks are the following 4-tuples: 0123 0124 0156 0257 0345 0367 0467 1267 1346 1357 1457 2347 2356 2456. The unique (7,3,1)-design is symmetric and has 7 blocks with each element repeated 3 times. Using the symbols 0 − 6, the blocks are the following triples: 013 026 045 124 156 235 346. This design is associated with the Fano plane, with the elements and blocks of the design corresponding to the points and lines of the plane. Its corresponding incidence matrix can also