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  "body": "\n## Summary\n\nA combinatorial design is a finite set together with a family of subsets, called blocks, that satisfies a balance condition.\nA typical condition is that every pair of points lies in the same number of blocks.\nDesign theory studies when such systems exist and how to build them.\nThe main families include [[block designs|4JCSS2Z1BA]], [[Steiner systems|7BPSC4T4M3]], [[finite projective planes|2HWB7P1DZN]] and, in a wider sense, [[Latin squares|CDVRMVED2V]].\nDesigns are used in the planning of experiments, in coding theory and in finite geometry.\n\n## Explanations\n\n### L1 human\n\nSuppose you want to test $7$ products in groups of $3$ so that every two products are compared in exactly one group.\nThis is possible with $7$ groups: $\\set{1,2,4}$, $\\set{2,3,5}$, $\\set{3,4,6}$, $\\set{4,5,7}$, $\\set{5,6,1}$, $\\set{6,7,2}$, $\\set{7,1,3}$.\nSuch a fair arrangement is a combinatorial design; this one is the Fano plane.\n",
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