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I know we can use inclusion-exclusion principle or stirling numbers to solve this for a set of n elements onto a set of m elements. But I wanted to know how can we get the result using simple combinatorics as the number of elements here is too less.

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    Do you mean the numbers $3$ and $2$ are too small to evaluate [$2!\,\genfrac\{\}0{}32$](http://en.wikipedia.org/wiki/Twelvefold_way#Surjective_functions_from_N_to_X)? Curious.2012-08-29
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    Inclusion-exclusion _is_ simple combinatorics, isn't it?2012-08-29

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