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quiz5solutions - Quiz 5 Solutions Dilip Raghavan December...

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Unformatted text preview: Quiz 5 Solutions Dilip Raghavan December 6, 2008 Problem 1. Let U be the set of all possible arrangements of the 26 letters without any repetitions. Let A 1 be the set of arrangements with the word INCH. Let A 2 be the set of arrangements with the word LOST, and let A 3 be the set of arrangements with the word THIN. We want to count A 1 A 2 A 3 . First of all, |U| = 26!. Now for | A 1 | , we treat INCH as one symbol, and so we have 23 symbols to arrange. So | A 1 | = 23!. The same for | A 2 | and | A 3 | . For | A 1 A 2 | , we treat INCH as one symbol and LOST as another symbol, and so we are arranging 20 symbols. So | A 1 A 2 | = 20!. Next, for | A 2 A 3 | , since there must be no repetitions, the only possibility is for LOSTHIN to be one symbol. So again we are arranging 20 symbols, and so | A 2 A 3 | = 20!. For | A 3 A 1 | , notice that it is impossible to have both THIN and INCH occurring because some letter will be repeated. So | A 3 A 1 | =...
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