Proving that ${n\choose k} = \frac{n - k + 1}{k} {n\choose k-1}$

Question

I'm having a hard time seeing how the following works:

$${n\choose k} =\frac{n - k + 1}{k} {n\choose k-1}$$

I've browsed through the web, but have not found any satisfying evidence as of now. Are there any geniuses out there that could go through the whole process of how we can reach this conclusion?

Answer 1

$${n\choose k} = \frac{n!}{k!(n-k)!}$$ $$\frac{n - k + 1}{k} {n\choose k-1} = \frac{n - k + 1}{k} \frac{n!}{(k-1)!(n-(k-1))!}$$ $$=\frac{n - k + 1}{k} \frac{n!}{(k-1)!(n-k+1))!}$$ $$= \frac{n!}{k!(n-k)!}$$ $$= {n\choose k}$$

Answer 2

You can ask your self: On how many ways can we chose $k$ people and a president among choosen?

First answer is ${n\choose k}\cdot {k\choose 1}$.

Second answer. We can first chose $k-1$ people and among rest a president. so that is ${n\choose k-1}\cdot {n-k+1\choose 1}$.