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  "body": "\n## Definition\n\nLet $t, v, k, \\lambda$ be positive integers with $t \\le k \\le v$.\nLet $X$ be a [[finite set|070R982G4B]], whose elements are called points.\nLet $\\mathcal{B}$ be a finite family of subsets of $X$, called blocks, with repetitions allowed.\nThe pair $(X, \\mathcal{B})$ is a *$t$-$(v, k, \\lambda)$ design* if:\n\n1. $\\abs{X} = v$.\n2. Every block has exactly $k$ points.\n3. Every subset of $X$ with $t$ points is contained in exactly $\\lambda$ blocks.\n\nA *combinatorial design* in the usual sense is a $t$-design of this kind.\nMore broadly, the term covers any finite set with a family of subsets that satisfies a prescribed balance condition of this sort.\n\n## Remarks\n\nA $2$-$(v, k, \\lambda)$ design is a balanced incomplete [[block design|4JCSS2Z1BA]], and a $t$-$(v, k, 1)$ design is a [[Steiner system|7BPSC4T4M3]] $S(t, k, v)$.\nCounting gives the number of blocks: $\\abs{\\mathcal{B}} = \\lambda \\binom{v}{t} / \\binom{k}{t}$.\n\nExample: $X = \\set{1, \\dots, 7}$ with blocks $\\set{1,2,4}$, $\\set{2,3,5}$, $\\set{3,4,6}$, $\\set{4,5,7}$, $\\set{5,6,1}$, $\\set{6,7,2}$, $\\set{7,1,3}$ is a $2$-$(7, 3, 1)$ design, the Fano plane.\nThe blocks are the translates of $\\set{1, 2, 4}$ modulo $7$, whose pairwise differences $\\pm 1, \\pm 2, \\pm 3$ cover each nonzero residue exactly once, so every pair of points lies in exactly one block.\n\nNon-example: $X = \\set{1, 2, 3, 4}$ with blocks $\\set{1, 2}$ and $\\set{3, 4}$ is not a $2$-$(4, 2, 1)$ design, because the pair $\\set{1, 3}$ lies in no block.\n",
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