John computes the sum of the elements of each of the 15 two-element subsets of
\(\{1,2,3,4,5,6\} \).
What is the sum of these 15 sums?
15 two-element subsets:
\(\begin{array}{|llllll|} \hline &1,& 2,& 3,& 4,& 5,& 6 \\ \hline &1,2 & 2,3 & 3,4 & 4,5 & 5,6 \\ &1,3 & 2,4 & 3,5 & 4,6 & \\ &1,4 & 2,5 & 3,6 & \\ &1,5 & 2,6 & \\ &1,6 \\ \hline \end{array}\)
\(\begin{array}{|lcll|} \hline \text{the numbers are: } && 5\times 1 \\ &+& 5\times 2 \\ &+& 5\times 3 \\ &+& 5\times 4 \\ &+& 5\times 5 \\ &+& 5\times 6 \\ \hline &=& 5\times (1+2+3+4+5+6) \\ &=& 5\times \left(\frac{1+6}{2}\right)\times 6 \\ &=& 5\times \left(\frac{7}{2}\right)\times 6 \\ &=& 105 \\\\ &=& (n-1)\times \left(\frac{1+n}{2}\right)\times n \\ &=& 3\times \binom{n+1}{3} \\ \hline \end{array} \)