Twelvefold way

In combinatorics, the twelvefold way is a name given to a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number. The idea of the classification is credited to Gian-Carlo Rota, and the name was suggested by Joel Spencer.

Let ${\displaystyle N}$ and ${\displaystyle X}$ be finite sets. Let ${\displaystyle n=|N|}$ and ${\displaystyle x=|X|}$ be the cardinality of the sets. Thus ${\displaystyle N}$ is an ${\displaystyle n}$-set, and ${\displaystyle X}$ is an ${\displaystyle x}$-set.

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