Editing Inclusion–exclusion principle (section)

===Number of onto functions===
Given finite sets ''A'' and ''B'', how many [[surjective function]]s (onto functions) are there from ''A'' to ''B''? [[Without any loss of generality]] we may take ''A'' = {1, ..., ''k''} and ''B'' = {1, ..., ''n''}, since only the cardinalities of the sets matter. By using ''S'' as the set of all [[Function (mathematics)|functions]] from ''A'' to ''B'', and defining, for each ''i'' in ''B'', the property ''P<sub>i</sub>'' as "the function misses the element ''i'' in ''B''" (''i'' is not in the [[Image (mathematics)|image]] of the function), the principle of inclusion&ndash;exclusion gives the number of onto functions between ''A'' and ''B'' as:<ref>{{harvnb|Mazur|2010|loc=pp.84-5, 90}}</ref>

:<math>\sum_{j=0}^{n} \binom{n}{j} (-1)^j (n-j)^k.</math>