If Santa Claus were to distribute presents randomly

On the day before Christmas, Santa Claus is making final preparations before flying out to deliver the presents. However, he discovers that the reindeer have eaten the list detailing which present should go to which child. Silly Santa, accidentally dropping the list in the reindeer stables!

Luckily he still has the list of all the good children. He also recalls that there was exactly one present for each of the $100,000,000$ good children. With Christmas Eve fast approaching, Santa sees no choice but to distribute the presents at random, one to each good child. Let $p$ be the probability that at least one child will get the gift that was meant for them in the first place. Find $\lfloor 1000p \rfloor$ .

(This problem is not entirely original)

$\underline{Assumptions:}$

• Santa Claus is real

• "Good children" are defined to be children who have been good this year, and who are not too old for the above assumption. In this problem, assume there are 100 million of them and 100 million presents.

• Each good child was meant to receive exactly 1 present in the first place, because Santa decided to be fair this year

• Santa randomly distributes exactly 1 present to each good child.

• Please ignore the fact that this problem was posted nowhere near Christmas.