Let A be a set of 8 distinct positive integers each 15 Show

Let A be a set of 8 distinct positive integers each 15. Show that

there are two subsets of A, both of size 2, that have the same sum.

Solution

So numbers can vary from 1 to 15 and are distinct.

So the sums vary from 1+2=3 to 14+15=29

So,27 possible sums varying from 3 to 29

Number of possible 2 element subsets of A are

C(8,2)=28

28 subsets but only 27 possible sums so by pigeonhole princple at least two subsets must have the same sum.