Summation proof (with binomial coefficents) I am trying to prove that $\sum_{k=2}^n$ $k(k-1) {n \choose k}$=$n(n-1)2^{n-2}$.
I was initially trying to use induction, but I think a more simple proof can be done using the fact that $\sum_{k=0}^n {n \choose k}$=$2^n$.
This is how I begin to proceed:
$\sum_{k=2}^n$ $k(k-1) {n \choose k}$= $\sum_{k=0}^{n-2}$ $(k+2)(k+1) {n \choose k+2}$= $2^{n-2} *\sum_{k=0}^{n-2}$ $(k+2)(k+1)$. 
First of all, is this correct so far? And second, how would I proceed from here. 

$$k(k-1)\cdot{n \choose k}=n(n-1)\cdot{n-2 \choose k-2}$$
