# the number of surjections that can be defined from A={1,2,8,9} to b={3,4,5,10} is?

A surjection from a set A of size n to a set B of size k may be characterized by a partition of A into k subsets, together with an permutation of the k elements of B.

The partitions are counted by the Stirling numbers of the second kind S(n,k), and the permutations are counted by k!, so there are

$S(n,k)k!=\sum _{i=0}^{k}(-1{)}^{i}\left(\underset{i}{\overset{k}{}}\right)(k-i{)}^{n}\phantom{\rule{0ex}{0ex}}$

In this case n=4,k=4

and from table for stirling numbers S(4,4)=1

In this case, that’s S(4,4)4!=1⋅24=24.

Regards

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