Just think of the integers as different numbers that get grouped into pairs. Two groups are considered "mates" if the same number belongs to both groups and "non-mates" if the two groups have no overlap.

With that, rephrase the question.

Q1. For any given pair, how many other pairs will have no overlap? (use #s 1 through 7 only)

Suppose the given pair is (1,2). This question simply becomes how many other pairs can we make form the 5 unused numbers? In other words, how many ways are there to pick 2 from 5? This is 5!(2!3!) = 10.

Q2. Given two pairs that share a common number, how many other pairs will have that number? (use #s 1 through 9 only)

Suppose the initial mates are (1,2) and (1,3). there are only six other pairs that will be mates to both original mates: (1,4), (1,5), (1,6), (1,7), (1,8) and (1,9)