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There are 10 students in the class and 10 seats in the classroom, and each student sits in the same seat during every lesson. However, the teacher insists that no student sit in their "usual" seat during an exam. How many different seating arrangements are permissible during an exam?

I'm thinking the first student has a choice of 10-1 seats to choose. then the next student would have 9-1, then 8-1, 7-1, 6-1, 5-1, 4-1, 3-1, 2-1. However, that doesn't seem to work because what if the last chair was the same seat the last student usually sits in.

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2 Answers 2

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Hint: You are looking for a derangement of the students

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  • $\begingroup$ This and that are identical to the bone. $\endgroup$
    – Asaf Karagila ♦
    Dec 2, 2013 at 23:55
  • $\begingroup$ @AsafKaragila: true. Should we call it a duplicate? $\endgroup$
    – Ross Millikan
    Dec 3, 2013 at 0:09
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Let $D_n$ be the number of derangements.
$$D_n=n!\sum_{i=0}^{n}\frac{(-1)^i}{i!}$$

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  • $\begingroup$ No. This would allow 1 and 2 to exchange places and all the rest to sit in their usual seats. $\endgroup$
    – Ross Millikan
    Dec 2, 2013 at 22:48
  • $\begingroup$ @RossMillikan, you're right. I'm going to edit my answer. Thx! $\endgroup$
    – EggHead
    Dec 16, 2013 at 19:48

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