How to prove that if n and k are integers with 1 ≤ k ≤ n, then k*(n C k)=n(n−1 C k−1) combinatorally? I am having with combinatorial proofs.  My professor says to come up with a scenario so that we can connect both sides by double counting but I am clueless.

There is a group of $n$ people. We want to give $k$ people in the group medals, one of them gold, and the others plastic. How many ways can we do this? Here are two ways of counting:
(i) We can choose the $k$ medal winners, and then from these choose the one who will get gold.
(ii) Or else we can choose the gold medal winner, and then the people who will get plastic.
