\begin{align*} _nP_r & = n(n-1) ... (n-r+1)\\ & = n(n-1) ... (n-r+1)\frac{(n-r)!}{(n-r)!}\\ & = \frac{n(n-1) ... (n-r+1)(n-r)!}{(n-r)!}\\ & = \frac{n!}{(n-r)!} \end{align*}