midterm2sol

Ways to permute the letters and 3 accounts for over

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Unformatted text preview: n: 7! . There are 7! ways to permute the letters, and 3! accounts for over counting for the 3! A’s. (d) (2 points) In an n dimensional hypercube, how many vertices are there at a distance of exactly 6 from a particular vertex? ￿￿ Solution: n . The vertex can be represented by an n-bit string and all vertices at a distance of 6 exactly 6 differ in 6 bits - we just need to choose which 6 bits. (e) (2 points) If A, B, C are event such that P[A] = .5, P[B] = .4 and P[C] = .3, and such that P[A ∩ B] = .2, P[A ∩ C] = .1, P[B ∩ C] = .1 and P[A ∪ B ∪ C] = .9. What is P[A ∩ B ∩ C]? Solution: .1. By the inclusion/exclusion principle, Pr[A ∪ B ∪ C] = Pr[A] + Pr[B] + Pr[C] − Pr[A ∩ B] − Pr[A ∩ C] − Pr[B ∩ C] + Pr[A ∩ B ∩ C]. (f) (5 points) Pick two non-zero numbers x and y modulo 7 at random. Let z = xy mod 7. Let A be the event that x = 3, B that y = 3 and C that z = 3. Note: An alternate interpretation would be picking a random non-zero number and reducing it modulo 7, so you are picking from the set {0, . . . , 6}. This was also given full credit, and the solutions for this interpretation (in italics) are included below. Of course you used the alternative interpretation the solution to part v. wa...
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This document was uploaded on 01/28/2014.

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