Combinatorial Interpretation

First, recall $\left(\begin{array}{c}n\\ k\end{array}\right)$ is the number of possible $k$ things chosen from an $n$ things.

Then, this formula state that by specifying the $k$ chosen things out of $n$, we are in effect specifying the $n-k$ unchosen things.

　Proof　：

$$\begin{array}{rcl}\left(\begin{array}{c}n\\ k\end{array}\right)& =& \frac{n\cdot (n-1)\cdots (n-k+1)}{k!}=\frac{n!}{k!(n-k)!}\\ & =& \frac{n!}{\{n-(n-k)\}!(n-k)!}=\frac{n!}{(n-k)!\{n-(n-k)\}!}\\ & =& \left(\begin{array}{c}n\\ n-k\end{array}\right)\end{array}$$