Editing Tuple (section)

=={{anchor|n-tuple}}{{math|''n''}}-tuples of {{math|''m''}}-sets ==

In [[discrete mathematics]], especially [[combinatorics]] and finite [[probability theory]], {{math|''n''}}-tuples arise in the context of various counting problems and are treated more informally as ordered lists of length {{math|''n''}}.<ref>{{harvnb|D'Angelo|West|2000|p=9}}</ref> {{math|''n''}}-tuples whose entries come from a set of {{math|''m''}} elements are also called ''arrangements with repetition'', ''[[Permutation#Permutations_of_multisets|permutations of a multiset]]'' and, in some non-English literature, ''variations with repetition''. The number of {{math|''n''}}-tuples of an {{math|''m''}}-set is {{math|''m''<sup>''n''</sup>}}. This follows from the combinatorial [[rule of product]].<ref>{{harvnb|D'Angelo|West|2000|p=101}}</ref> If {{math|''S''}} is a finite set of [[cardinality]] {{math|''m''}}, this number is the cardinality of the {{math|''n''}}-fold [[Cartesian product#n-ary Cartesian power|Cartesian power]] {{math|''S'' × ''S'' × ⋯ × ''S''}}. Tuples are elements of this product set.