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Unformatted text preview: 21127, Concepts of Mathematics, Review ProblemsPlease note that the Principle of InclusionExclusion is NOT on this exam.1. Prove that for everyk1 there exists some powernsuch that 7nhasktrailing zerosfollowed by a 1. For example, fork= 2 we have 720= 79792266297612001. (Hint:there are an infinite number of powers of seven, but only a finite number of possibleremainders).2. A derrangement, denoteddn, is a permutation of [n] such that no point is fixed. Forexample, 231 and 312 are the only derrangements of [3]. Use the principle of inclusionexclusion to find an expression fordn.3. Prove that the probability that a permutation ofNbeing a derrangement is exactly1e,where a permutation ofNis a permutation of all the natural numbers.4. Find the number of eightletter words using letters from the set{a,b,c}, if each lettermust occur at least once in each word.5. Find the number of quadratics of the formx2+ax+bthat do not have a root modulopwherepis any prime. (Hint: use the principle of inclusionexclusion withis any prime....
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 Spring '07
 GHEORGHICIUC
 Math

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