\(5^{10} - \left[{5 \choose 1}4^{10} - {5 \choose 2}3^{10} + {5 \choose 3}2^{10} - {5 \choose 4}1^{10}\right]\) functions. The \(5^{10}\) is all the functions from \(A\) to \(B\text{.}\) We subtract those that aren’t surjective. Pick one of the five elements in \(B\) to not have in the range (in \({5 \choose 1}\) ways) and count all those functions (\(4^{10}\)). But this overcounts the functions where two elements from \(B\) are excluded from the range, so subtract those. And so on, using PIE.
in-context