There are \(2^5\) functions all together, two choices for where to send each of the 5 elements of the domain. Now of these, the functions which are not surjective must exclude one or more elements of the codomain from the range. So first, consider functions for which \(a\) is not in the range. This can only happen one way: everything gets sent to \(b\text{.}\) Alternatively, we could exclude \(b\) from the range. Then everything gets sent to \(a\text{,}\) so there is only one function like this. These are the only ways in which a function could not be surjective (no function excludes both \(a\) and \(b\) from the range) so there are exactly \(2^5 - 2\) surjective functions.
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