Problem
A group of 50 people are comparing their birthdays (as usual, assume their birthdays are independent, are not February 29, etc.). Find the expected number of days in the year on which at least two of these people were born.
Solution
By linearity of expectation, the answer is 365 times the probability that at least two people were born on a given day. For a given day, there are 50 choose 2 or 1225 ways to choose two people who are born on that day and since the remaining people could be born on any day there are 365^48 choices for their birthdays. Dividing by 365^50, the number of possibilities with no restrictions, and multiplying by 365 yields, the expected number of days on which at least 2 people were born is 365(1225*365^48/365^50) = 1225/365, which is incorrect.
What is wrong about this approach?