quiz3_solution

quiz3_solution - Question 1 (a) This problem is just...

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Question 1 (a) This problem is just finding D 5 , the derangement of 5 people (the seats are distinguishable). According the formula in the notes, D 5 = 44. You can find it directly by applying inclusion- exclusion principle, or just use the formula D 5 = 5 ! 1 1 1 ! 1 2 ! 1 3 ! 1 4 ! 1 5 ! . (b) Let the 5 people be 1, 2, 3, 4 and 5. Denote A i be the event that the person sitting on the left of person i in the first meeting is same as that in the second meeting. Then we are finding A 1 A 2 A 3 A 4 A 5 . By inclusion exclusion principle, we know: A 1 A 2 A 3 A 4 A 5 ∣=∣ N ∣− A j ∣ A j A k ∣− A j A k A m ∣ A j A k A m A r −∣ A j A k A m A r A s Note that the seats are NOT distinguishable this time, so we are considering circular permutation. Hence | N | = 24 only. The value of each term in the the first, second, third and the fourth summands are 6, 2, 1 and 1 resp. So The answer is
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quiz3_solution - Question 1 (a) This problem is just...

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