$\sum_{k=1}^m k(k-1){m\choose k} = m(m-1) 2^{m-2}$ Thanks for those who helped me yesterday. I have another one and I'd appreciate if you can shed some light upon to keep me going. This one is related with binomial theorem. 
$$\sum_{k=1}^m k(k-1){m\choose k} = m(m-1) 2^{m-2}$$
Thanks for your thoughts!

Both sides are all ways of choosing a subset of $m$ things, with $2$ distinguished things in the subset (forcing the size of the subset to be at least $2$). The left side is conditioning on the size of the subset.
