I'll approach this answer assuming you have a calculator with an nCr button (we have 5 members in the set; if we want to know how many unique sets with 2 numbers per set then we enter for nCr 5C2 (5 choose 2). 5C2= 10.
This tells us there are 10 combinations of 2 in a set of 5. In terms of position, the positions of the numbers are position 1,2,3,4,5. Combinations of 2 include 1-2, 1-3, 1-4, 1-5, 2-3, 2-4, 2-5, 3-4, 3-5, 4-5; or 10 pairs.
To find the total number of subsets you need to find the number of subsets with 1 element, the number of subsets with 2 elements, ... 3 elements, 4 elements and 5 elements.
The total nuber of subsets of a 5 element set is
5C1 + 5C2 +5C3 + 5C4 +5C5 ; you normally add one more set to the answer to represent the empty set.
5C2 = 10
5C4 = 5
The total number of subsets is 32, or 33 if you count the empty set