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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 $2!\,\genfrac\{\}0{}32$? Curious. – Marc van Leeuwen Aug 29 '12 at 10:41
Inclusion-exclusion is simple combinatorics, isn't it? – Henning Makholm Aug 29 '12 at 12:20

You know probably the number off all functions from $\lbrace 1,2,3 \rbrace$ to $\lbrace 1,2\rbrace$ and a function which is NOT surjective must be constant since the range has only two elements.

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