<s>[INST] 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?
 [/INST] $$\sum_{r=2}^n \binom{n}{r} \binom{r}{2}
=\sum_{r=2}^n \binom{n}{2} \binom{n-2}{r-2}
=\binom{n}{2} \sum_{r=2}^n \binom{n-2}{r-2}
=\binom{n}{2} 2^{n-2}
=n(n-1) 2^{n-3}
$$
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