How can I prove that the probability that exactly 2 people share the same birthday is more likely than everyone has a different day out of 20 people?

Question

I will be thankful if you can help me and show how to solve this.

Answer 1

Hint

You can't prove it because it isn't true.
However, if you want to compute the probability that out of $20$ people, exactly two share the same birthday,

• There are $\large\binom{20}2$ ways the pair can be chosen, and $365$ ways they can share a birthday (ignoring leap years)
• The remaining $18$ have $364.363.362.361...347$ ways of having distinct birthdays different from the pair
• There are $365^{20}$ ways of having unconstrained birthdays
• You will find the probability to be well below $0.5$

Can you now compute what the number of people should be there for your assertion to be true ?