How many ordered pairs of sets (A, B) have the properties:
1. A is a subset of {1, 2, 3, 4, 5, 6};
2. B is a subset of {2, 3, 4, 5, 6, 7, 8};
3. the intersection of A and B has exactly 3 elements.

Source: Purple Comet Math Meet 2011: Middle School Level

Level: Junior

Let S = {1, 2, 3, 4, 5, 6} and T = {2, 3, 4, 5, 6, 7, 8}. The number 1 is in S only; the numbers 7 and 8 are in T only; and the numbers 2, 3, 4, 5, and 6 are in both S and T. We can put 1 in A or not in A in 2 ways. We can put each of 7 and 8 in B or not in B in 2 ways. There are 5C3 = 5*4*3/3! = 10 ways to select 3 numbers from 2, 3, 4, 5, and 6 to put in both A and B. For each of the remaining 2 numbers not being put in both A and B, it belongs to A only, B only, or neither A nor B in 3 ways. In the end, the number of (A, B) is 2*(2^2)*10*(3^2) = 720.