\[ m^n - \sum_{k=1}^{m} (-1)^{k+1} {m \choose k} (m-k)^n
= m^n + \sum_{k=1}^{m} (-1)^{k} {m \choose k} (m-k)^n
= \sum_{k=0}^{m} (-1)^{k} {m \choose k} (m-k)^n
\] % \end{eqnarray*}