Combinatorial Proof

Say we want to choose a baseball team of $k$ players from $n$ players and we want to choose a captain for the baseball team. We can count the above in two different ways.

$1$. Choose the captain first. This can be done in $n$ ways. Now choose the rest of the team, i.e., we need to choose $k-1$ people from the remaining $n-1$ people, which can be done in $\dbinom{n-1}{k-1}$ ways.

$2$. Choose the team first, i.e., choose $k$ players from $n$ players. This can be done in $\dbinom{n}k$ ways. Now once we have these $k$ players, the captain can be chosen in $k$ ways.

We have a group of $n$ people, and want to count the number of ways to choose a committee of $k$ people with Chair.

For the left-hand side, we select the Chair first, and then $k-1$ from the remaining $n-1$ to join her.

For the right-hand side, we choose $k$ people, and select one of them to be Chair.