Problem corner

Suppose that there are n people in a room. What is the exact probability that there are exactly two people (A,B) such that A and B have the same birthday and no others share a birthday? Note that this is a variation of the famous birthday paradox where the question is to determine the probability that at least two of n people have the same birthday. Determine the probability for n=5 as an exact fraction. Give a decimal approximation for n=30. Then give a formula for the general case for any n>2. Challenge: Extend the problem for two distinct pairs (A,B), (C,D) such that A,B share a birthday, and C,D share a different birthday and everyone else has a distinct birthday.