It doesn't need a small number -- rather it relies on you being able to find a pairing amongst any of your candidates, rather than find a pairing for a specific birthday.
That's the paradoxical part: the number of potential pairings for a very small number of people is much higher than one might think, and so for 365 options (in the birthday example) you can get even chances with far fewer than 365, and even far fewer than ½x365 people..
That's the paradoxical part: the number of potential pairings for a very small number of people is much higher than one might think, and so for 365 options (in the birthday example) you can get even chances with far fewer than 365, and even far fewer than ½x365 people..