The problem is a variation of Set Cover(

http://en.wikipedia.org/wiki/Set_cover_problem) whose optimization is NP-hard:

Take the set S of combinations of 5 numbers from 1-10, which will have 252 elements in total.

Take a new set of subsets of S where each subset contains all reachable 5-combinations from each 3-combination, and call this set T.

For example, an element in T associated with {1,2,3} would look like:

{ {1,2,3,4,5}, {1,2,3,4,6}, ..., {1,2,3,5,6}, ..., {1,2,3,9,10} }, which has 21 elements.

Then the problem becomes to choose minimal number of sets in T that will cover all elements in S.

I think the answer (20) you got is quite reasonable, though I didn't try to prove or verify it yet.