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?



Answer :

Sum of the 15 two elements subset of $\{1,2,3,4,5,6} = ( N-1) (N)(N+1)/2

Three elements....two at  time  (1,2) (1,3) (2,3)....sum  =   12

Four elements .... two at a time  (1,2) (1,3) (1,4) (2,3) (2,4) (3,4)  .....sum  =  30

Five elements....two at a time  (1,2) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5)

Note that for N elements taken 2 at a time....the pattern of sums seems to be

= (N-1)(N)(N+1) / 2

For taking 6 elements taking 2 at a time the sum should be

=(5)(6)(7) /2

=210 /2

=105

Therefore, total sum  of those 15 sums is = 105

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