Determining the closed form of $\sum\limits_{r = 2}^n \binom{n}{r} \binom{r}{2}$ Here is a sum in combinatorics, $\sum\limits_{r = 2}^n \binom{n}{r} \binom{r}{2}$ where $n>2$, does this have a closed form?

Yes, the closed form is
$$\sum\limits_{r = 2}^n {C_n^rC_r^2} = n \cdot (n-1) \cdot 2^{n-3}$$
Seems there're various ways to prove this.
The combinatorial approach to prove this would be to think about this.
In how many ways can we choose from $n$ people a committee with $r$ members (where $ 2 \le r \le n$), and also choose $2$ chairmen from the committee. Express this number in two different ways and you get the above identity.
