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quizsol - N distinct objects by binary strings of length k...

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CS 346 Cryptography FALL 2009 Background Quiz - Solutions 1. What is the numeric value of log 2 (8 50 )? 150 2. Give a big-O estimate for each of the following functions: A. 2 n 3 + n 2 log n = O ( n 3 ) B. ( n 2 + 8)( n + 1) = O ( n 3 ) C. n ! + 2 n = O ( n !) 3. An integer is chosen at random from the first 300 positive integers. What is the probability that the integer chosen is divisible by 2 or by 3? Let E denote the event that the integer chosen is divisible by 2, and let F denote the event that the integer chosen is divisible by 3. Then P ( E ) = 150 / 300 = 1 / 2 and P ( F ) = 100 / 300 = 1 / 3. We are looking for the probability P ( E F ). We know that P ( E F ) = P ( E ) + P ( F ) - P ( E F ). P ( E F ) = 50 / 300 = 1 / 6, thus P ( E F ) = 1 / 2 + 1 / 3 - 1 / 6 = 2 / 3. 4. What is the greatest common divisor of the following pair of integers: 2 2 · 7 and 5 3 · 13? 1 5. What is the number of different binary strings of length
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Unformatted text preview: N distinct objects by binary strings of length k , such that different objects are represented by different strings. How large k has to be? d log 2 N e 7. What is the number of different subsets of size k of an n element set? ± n k ² 8. Prove that ∑ n i =0 ± n i ² = 2 n . (HINT: there is a simple argument, without formal calculations.) We have seen before that the number of different binary strings of length n is 2 n . We also know that the number of different binary strings that have exactly i 1’s is ± n i ² . (Compare this with question 7.) Thus the number of different strings of length n is ∑ n i =0 ± n i ² . This proves that ∑ n i =0 ± n i ² = 2 n . 2...
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