Actually there are possible subsets A and the maximum sum of elements of A is 189, but those aren't the numbers we're interested in.

The minimum sum of elements of a subset of cardinality 5 here is 1+...+5 = 15. The maximum is 21+...+25 = 115. There are a total of 101 possible values for such a sum, then. There are possible 5-subsets of any given A with cardinality 9. Thus we apply the pidgeonhole principle noting that 126 > 101.