Well notice that the set A + B is neither included in the set A nor in the set B. Therefore the number of combinations are 5 (the number of elements in A) + 7 (the number of elements in B) = 12.

That is because the smallest element in A is 15, and in B it is 8. Yet, 15 + 8 = 23 which is greater than the greatest element in A or B. Do it the other way too, and you have a proof (I think ).