In a room with 23 people, there is a 50% chance that two of the 23 share a birthday. In a room of 75 people, it jumps to a 99.9% chance. It is hard for us to comprehend that level of compounding power of exponents.
It’s just math, though. In a room of 23 people, you think of yourself compared to the other 22 people. We forget that all the other people are comparing themselves with everyone else, too.
This raises the comparisons to 231 comparisons and possible matches. There are ten times as many comparisons that don’t include yourself. This explains how the paradox can happen.
The easiest way to think about it is not by thinking of how many matches one can kind. Finding the chance that everyone is different is the easiest way. If you take 23 people multiplied by 22 people and then divide by 2 (the least amount of people to share a birthday in the group) then you come up with the number 253.