How can i prove this identity using the combinatorial method?

Question

$$m\,C(n,m)=n\,C(n-1,m-1)$$ I tried something like how can we choose a group out of n people and then how many ways we can choose the leader , and it worked for the first side but not for the 2nd. Thank you in advance

Answer 1

You have $n$ people and you are choosing from them team of $m$ players with one captain.

Left-hand side: You pick $m$ players and $C(n,m)$ and then you pick one captain among them.

Right-hand side: You pick a captain and then out of the remaining $n-1$ guys you choose rest of the team $C(n-1,m-1)$.