For example, the pair represents the case where persons 1,2,3, & 4 get ice-cream cones while persons 2,4,6, & 8 each gets a chocolate-chip cookie.
If A is a non-empty subset of the eight, we cannot have the pairs because we are given “that at least one person gets both an ice-cream cone and a chocolate-chip cookie”. Likewise we cannot have the pair because of the given.
There are subsets of a set of eight. There are ordered pairs of those subsets. Now remove the cases we cannot have.
Can you finish?