Let $S(n)$ be the set of permutations of $\{1,2,...,n\}$, and assume $a_i$ is a real number for $i=1,2,...,n$. Why is it true that $$\sum_{\sigma \in S(n)} a_{\sigma(1)}a_{\sigma(2)} = 2(n-2)! \sum_{1\le i<j \le n}a_ja_k?$$
Similarly, we also have $$\sum_{\sigma \in S(n)} a_{\sigma(1)} = (n-1)! \sum_{j=1}^na_j.$$