What is the combinatorial proof for the formula of Stirling numbers of the second kind ?
$${n\brace k}=\frac1{k!}\sum_{j=0}^k(-1)^{k-j}\binom{k}jj^n$$
where ${n\brace k} = S\left(n,k\right)$ is the number of set partitions of a fixed $n$-element set into $k$ parts.
Count non-surjective functions $[n]\to [k]$. Use inclusion-exclusion by counting functions $A_i$ that miss one particular element $i\in [k]$ and then consider $A_1\cup A_2 \cup \dots \cup A_k$.