Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$

Suppose we want to prove $$ k \binom{n}{k} = n \binom{n-1}{k-1}$$

In the LHS we are choosing a team of $k$ players from $n$ players. Then we are choosing a captain. In the RHS we are choosing a captain from the $n$ players. Then we are choosing the remaining $k-1$ players from the $n-1$ players.

Is this a correct interpretation?

This question is Identity 130 on page 65 of "Proofs that Really Count" by Benjamin and Quinn.

Question: How many ways can we create a size $k$ committee of students from a class of $n$ students, where one of the committee members is designated as chair?

Answer 1: There are $\binom{n}{k}$ ways to choose the committee, then $k$ ways to select the chair. Hence there are $k\binom{n}{k}$ possible outcomes.

Answer 2: First select the chair from the class of $n$ students. Then from the remaining $n-1$ students, pick the remaining $k-1$ committee members. This can be done $n\binom{n-1}{k-1}$ ways.