MIT6_042JF10_assn09 - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science November 2, 2010 Tom Leighton and Marten van Dijk Problem Set 9 Problem 1. [10 points] (a) [5 pts] Show that of any n + 1 distinct numbers chosen from the set { 1 , 2 ,..., 2 n } , at least 2 must be relatively prime. ( Hint: gcd( k,k + 1) = 1 . ) (b) [5 pts] Show that any Fnite connected undirected graph with n 2 vertices must have 2 vertices with the same degree. Problem 2. [10 points] Under Siege! ±earing retribution for the many long hours his students spent completing problem sets, Prof. Leighton decides to convert his office into a reinforced bunker. His only remaining task is to set the 10-digit numeric password on his door. Knowing the students are a clever bunch, he is not going to pick any passwords containing the forbidden consecutive sequences ”18062”, ”6042” or ”35876” (his MIT extension). How many 10-digit passwords can he pick that don’t contain forbidden sequences if each number 0 , 1 ,..., 9 can only be chosen
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MIT6_042JF10_assn09 - 6.042/18.062J Mathematics for...

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