A group of 100 people are comparing their birthdays (as usual, assume their birthdays are independent and not on February 29, etc.). Find the expected number of pairs of people with the same birthday, and the expected number of days in the year on which at least two of these people were born.

In the given problem, we have a group of 100 people, each with an independent birthday that is uniformly distributed over the 365 days of a year (excluding leap years). We need to find the expected number of pairs of people with the same birthday and the expected number of days on which at least two people were born.

1. Expected Number of Pairs with the Same Birthday

The first part of the problem asks for the expected number of pairs of people with the same birthday. To solve this, we will first compute the number of possible pairs in the group and then calculate the probability that a given pair has the same birthday.

The total number of distinct pairs of people that can be formed from the 100 people is given by the combination formula:

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