Combinatorics

# Then expanding the summa5on we have example

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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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