CO35 The Hat-Check Problem and Counting Derangements

If n patrons check their hat, and the hats are returned randomly, what is the probability that no one gets their own hat back? Does this probability increase or decrease if n -- the number of hats -- goes from 10 to 10,000?
See the answer to this question as well as formulas and recurrence relations for the number of derangements.

Videos on the Inclusion-Exclusion principle:
CO32 The Inclusion-Exclusion Principle CO32 The Inclusion-Exclusion Principle
CO34 Combinations of Multisets via Inclusion-Exclusion CO34 Combinations of Multisets via In...
CO35 Derangements CO35 The Hat-Check Problem and Counti...
CO36 Non-attacking Rooks on Boards with Forbidden Positions CO36 Non-attacking Rooks on Boards wi...

Part of a series of lectures on introductory Combinatorics. This full course is based on my book
Shahriar Shahriari, An Invitation to Combinatorics, Cambridge University Press, 2022.
DOI: https://doi.org/10.1017/9781108568708
For an annotated list of available videos see
https://pomona.box.com/s/by2ay2872avx...
YouTube Playlist: Combinatorics, An Invitation

For more undergraduate math videos subscribe to the channel  @Shahriar Shahriari

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College