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\begin{align*} P(n+1;k) &= (n+1)P(n;k-1) \\ &= (n+1)\frac{n!}{(n-(k-1))!} \\ &= \frac{(n+1)n!}{(n-k+1)!} \\ &= \frac{(n+1)!}{((n+1)-k)!} \end{align*}