TITLE: Combinatorial Proof
QUESTION [5 upvotes]: I have trouble coming up with combinatorial proofs. How would you justify this equality?
$$
n\binom {n-1}{k-1} = k \binom nk
$$
where $n$ is a positive integer and $k$ is an integer.

REPLY [8 votes]: 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.