quiz 5 solution - 22M:150:001 Quiz 5 Duration of quiz 10...

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22M:150:001 Quiz 5 Duration of quiz: 10 minutes. 24 th October 2012 Calculators, books or formula sheets are NOT allowed. Name: SOLUTIONS 1. Does a sequence of 9 distinct real numbers always have an increasing OR decreasing subsequence of 4 numbers? (5 points. ) Solution: No, a sequence of 9 distinct real numbers does not always have an increasing or decreasing subsequence of 4 numbers. (The pigeonhole principle guarantees that in any sequence of 10 distinct real numbers, we always have an increasing or decreasing sequence of 4 numbers.) To complete the solution, here is a string of 9 distinct natural numbers which does not have an increasing or decreasing subsequence of 4 numbers: 7 , 8 , 9 , 4 , 5 , 6 , 1 , 2 , 3 . 1
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2. Recall that a bit string is a string consisting of only the digits 0 and 1. How many bit strings of length 12 contain: a) exactly three 1’s? b) at most three 1’s? c) an equal number of 0’s and 1’s? (5 points. ) Solution: a) We want to count the number of bit strings of length 12 consisting of exactly three 1’s i.e., we want to find the number of possible arrangments of nine 0’s and three 1’s in 12 positions.
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