Hello epi1988The number of 5-element subsets is the number of ways of selecting 5 items from 9 = .

The original set A = {1, 2, ..., 9} may be partitioned into 5 subsets: the four subsets comprising S-pairs: {1, 9}, {2, 8}, {3, 7} and {4, 6}, and the subset {5}. Suppose now that B is a 5-element subset that does not contain any S-pairs. Then B contains exactly one element from each of the 5 subsets in the above partition, since otherwise it will contain an S-pair. Therefore B contains the element 5.

To find the number of subsets containing no S-pairs: the element from each of the four S-pair subsets in this partition of A can be chosen in 2 ways, and the fifth element (5) in 1 way. So there are such subsets altogether.

These subsets are {1, 2, 3, 4, 5}, {9, 2, 3, 4, 5}, ... ?

I think you can complete it now.

Grandad