Let $n$ and $k$ be positive integers. Then binomial coefficients has symmetry property such as ：
$$\left(\begin{array}{c}n\\k\end{array}\right)=\left(\begin{array}{c}n\\n-k\end{array}\right)$$
and this is called the symmetry identity.
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{eqnarray*}\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{eqnarray*}