Prove that $$ {n\choose k}= \frac{n}{k} {n-1\choose k-1}$$
I have been thinking about it for a very long time and I have finally come to this solution:   

First of all, multiply by $k$:   

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

Let's say that we have a group of $n$ people. The left hand side first chooses $k$ of them who get a candy and then chooses $1$ from the chosen group to get a toy truck.   

The right hand side first chooses one person who gets a candy and a toy truck and then chooses $n-1$ people to get a candy.   

In both scenarios we have chosen exactly $k$ people, $k$ of them got a candy and $1$ of them got the toy truck. Thus, LHS and RHS are the same.   

Is this solution valid to any extent?   

What other ways to prove this are there? I tried induction - it works, too, but is a little tedious.