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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 (using inclusion-exclusion) and recurrence relations for the number of derangements.#combinatorics Subscribe @Shahriari 00:00 Introduction 00:13 The Hat Check Problem 01:42 Derangements 03:33 D_n, the number of derangements, for small n 04:44 A formula for D_n 05:35 The Inclusion-Exclusion principle ( • CO32 The Inclusion-Exclusion Principle ) 07:22 Proof of the formula using inclusion-exclusion 11:51 Example D_5 13:01 Alternative interpretation: partial sums of a series for 1/e 14:24 Theorem: D_n/n! converges to 1/e as n goes to infinity 15:10 Proof of Theorem 16:59 A recurrence relation: D_n = (n-1)(D_{n-1}+D_{n-2}) 20:46 A second recurrence relation: D_n = nD_{n-1} + (-1)^n 23:28 Wrap up Videos on the Inclusion-Exclusion principle: • CO32 The Inclusion-Exclusion Principle • CO34 Combinations of Multisets via Inclusi... • CO35 The Hat-Check Problem and Counting De... • CO36 Non-attacking Rooks on Boards with Fo... 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 Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award.