Consider the following counting problem. Among $n$ boys and $n$ girls, select equal numbers of boys and girls to form two teams and then select one from each to be the captain. How many different ways are there?

Let's assume each team has $k$ people. There are $\binom{n}{k}$ ways to select $k$ boys from $n$ candidates. Then there are $k$ ways to choose one to be the captain. Therefore, there are $k\binom{n}{k}$ ways to form the boy's team. Similarly, there are $k\binom{n}{k}$ ways to form a girl's team. Together, there are $k^2\binom{n}{k}^2$ ways to form $k$-people teams. When $k$ changes from $0$ to $n$, there are totally $$\sum_{k=0}^nk^2\binom{n}{k}^2$$

ways. On the other hand, we can first choose one boy and one girl to be captains. There are $n^2$ different ways. Then the total number of ways to choose equal number of boys and girls are $$\sum_{k=0}^{n-1}\binom{n-1}{k}^2 = \binom{2n-2}{n-1}$$

by Vandermonde's identity.

Because these are two different approaches to solve the same problem, these two quantities are equal.