Prove $\binom{n}{k}=\frac{n}{k}\binom{n-1}{k-1}$

Question

I want to make a combintronic proof for this. So far, I have we want to form a committee of k members with a president from n amount of people.

Thus on the left side we can select the k members from the n amount of people and then select a president from there.

How would I go about doing the Right side though?

Answer 1

Using the factorial definition of $\binom{n}{k}$, you get that $$\binom{n}{k}=\frac{n!}{k!\cdot(n-k)!}=\frac{n\cdot (n-1)!}{k\cdot (k-1)!\cdot(n-k)!}=\frac{n}{k}\binom{n-1}{k-1}.$$