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  "body": "\n## Summary\n\nThe twelvefold way is a table of twelve basic counting problems, see the [[definition|3BBE4XY4E9]].\nEach problem counts maps from a finite set $N$ to a finite set $X$.\nThe maps may be arbitrary, injective or surjective.\nThey may be counted as they are, or up to permutations of $N$, of $X$, or of both.\nThe twelve answers bring together powers, [[falling factorials|0SJNCJCCQG]], [[binomial coefficients|0GBZM0MT0F]], [[Stirling numbers of the second kind|0JCQGPY8Z4]] and [[integer partitions|0JV30XN73Z]].\n\n## Explanations\n\n### L1 human\n\nThink of $n$ balls that you put into $x$ boxes.\nThe balls may be labelled or identical, and the boxes may be labelled or identical.\nYou may also require at most one ball per box, or at least one ball per box.\nEach choice gives one counting problem, and there are $2 \\cdot 2 \\cdot 3 = 12$ of them.\nFor example, $3$ labelled balls go into $2$ labelled boxes in $2^3 = 8$ ways, but $3$ identical balls go into $2$ labelled boxes in only $4$ ways.\n",
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