When $k>0$, the identity can be rewrite as

$$\left(\begin{array}{c}n\\k\end{array}\right)=\frac{n}{k}\left(\begin{array}{c}n-1\\k-1\end{array}\right)$$

and we often use to absorb a variable into a binomial coefficient when that variable is a nuisance outside.

　Proof　：

\begin{eqnarray*}k\times \left(\begin{array}{c}n\\k\end{array}\right)&=&k\times\frac{n!}{k!(n-k)!}\\ &&\\&=&\frac{n!}{(k-1)!(n-k)!}\\ && \\&=&\frac{n(n-1)!}{(k-1)!(n-k)!}\\ && \\&=&n\times\frac{(n-1)!}{(k-1)!\{(n-1)-(k-1)\}!}\\ && \\&=&n\times \left(\begin{array}{c}n-1\\k-1\end{array}\right)\end{eqnarray*}