Unformatted text preview: ed an r permuta5on • Theorem: The number of r permuta5ons of a set of n dis5nct elements is • It follows that • In par5cular • Note here that the order is important. It is necessary to dis5nguish when the order maxers and it does not 235 28 CSCE Combinatorics Applica5on of PIE and Permuta5ons: Derangements (I) (Sec5on 7.6) • Consider the hat check problem – Given • An employee checks hats from n customers • However, s/he forgets to tag them • When customers check out their hats, they are given one at random – Ques5on • What is the probability that no one will get their hat back? CSCE 235 Combinatorics 29 Applica5on of PIE and Permuta5ons: Derangements (II) • The hat check problem can be modeled using derangements: permuta5ons of objects such that no element is in its original posi5on  Example: 21453 is a derangement of 12345 but 21543 is not • The number of derangements of a set with n elements is • Thus, the answer to the hatcheck problem is • Note that • Thus, the probability of the hatcheck problem converges See textbook, Sec+on...
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