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A group of ten people give their hats to the coatroom attendant. Five of the ten are wearing sombreros, and five and wearing fedoras. How many ways can the clerk return the hats so that no one gets their hat back if,

a: No one gets the right kind of hat

b: Everyone gets the right kind of hat

c: The attendant loses the sombreros and only returns the 5 fedoras. However the five people in the group that do get a hat back, may or may not have gotten the correct hat back.

dylan absect
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1 Answers1

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a. 5!*5! If the kind of hat is wrong, we just get permutations.

b. D5 * D5 (Where D5 is the number of derangements on 5 objects = [5!/e]) This derangement number is determined with the inclusion/exclusion method.

c. 10*9*8*7*6. Without the correct hat/person limitations, the attendant can give the five fedoras to 10, 9, 8, 7, 6 persons respectively (assuming every person gets one hat)

Pieter21
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  • Thanks for the response! However I'm thinking the answers would be something more like http://math.stackexchange.com/questions/627913/question-on-the-hat-check-problem – dylan absect Nov 28 '15 at 18:23
  • That will be in the next college ;) – Pieter21 Nov 28 '15 at 18:28
  • I'm confused with your response? The reason I referenced that link is because it looks a lot more like what I am doing in class vs the more simple answer you gave. I don't think just using factorials in taking into account the larger amount of variation and choices that can arise from the problem. – dylan absect Nov 28 '15 at 18:32
  • I'll explain the answers, though indeed b. is the answer where you do the inclusion/exclusion stuff, and D5 is the short answer. – Pieter21 Nov 28 '15 at 18:47
  • the added explanation was helpful, Thanks! – dylan absect Nov 28 '15 at 18:53