A question on counting I'm having difficulty answering (Qb iii)
Question:
(a) Write an expression for the number of sets S which contain 10 elements, each of which is
an integer between 1 and 20.
There are ${10 \choose 2}$ subsets of size 2.
(iii) For a non-empty subset X ⊆ S, let t(X) denote the sum of the members of X. Prove
that there must be distinct subsets A,B ⊆ S, each of size two, such that t(A) = t(B).
(Hint: what are the possible values of t(A) and t(B)?)
Where I am at so far
I realised the following:
Consider 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20.
I realised that if you pick any two number say 5 and 13 then 
5 + 13 = 18, 
6 + 12 = 18, 
7 + 11 = 18 etc.
But that's really all I've realised.
