203_W10_Ex3-solns

# 203_W10_Ex3-solns - EECS 203 Discrete Mathematics Winter...

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EECS 203, Discrete Mathematics Winter 2010, University of Michigan, Ann Arbor Solution to Exam 3 Prepared by Xiaodi Wu Section 1. Problem 1. Which, if any, of the following statements are true? Circle the true answers on page 2. (A) The number of 8-bit binary strings that contain exactly three 1’s is 24. Ans: False. The number should be ( 8 3 ) = 56. (B) The number of ASCII strings of Fve characters that contain the symbol # at least once is over one billion. (Note that there are 128 di±erent ASCII characters.) Ans: True. The number should be 128 5 127 5 . By using the fact that 128 5 = (127+1) 5 and its expansion we have 128 5 127 5 =5 × 127 4 + ··· where the Frst term is over one billion. (C) There are n ! / ( n m )! one-to-one functions from a set of cardinality m to one of cardinality n , for any integers m , n with 1 m n . Ans: True. Any permutation choosing m out of n will give one such function uniquely. Thus this number is n ! / ( n m )!. (D) At a party where there are n 2 people, there are at least two people who know the same number of other people at the party. Ans: True. Same problem in the homework. Problem 2. Which, if any, of the following statements are true? Circle the true answers on page 2. (A) The number of permutations of the eight letters ABCDE²GH that contain the string ABC is 720. Ans: True. We can treat ”ABC” as a single character. Then the number is 6! = 720. (B) The coeﬃcient of a 3 b 3 in the expansion of ( a + b ) 6 is 20. Ans: True. By deFnition, the coeﬃcient should be ( 6 3 ) = 20. (C) ²or any integers n , k with 0 k n , k ! always divides n ( n 1)( n 2) ··· ( n k +1). Ans: True. Given the fact that the ( n k ) = n ! k !( n k )! is always an integer, thus we have k ! | n ( n 1)( n 2) ··· ( n k +1) (D) The number of di±erent Fve-letter strings that can be made from all the letters in EULER is 20. Ans: False. ²ollow similar method in the sample exam, we have the number is 5! 2! = 60. 1

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Problem 3. Which, if any, of the following statements are true? Circle the true answers on page 2. (A) On throwing a fair dice twice, the probability that the two upturned numbers sum to 8 is 1 6 . Ans: False. All the pairs with summation of two outcomes being 8 are (2 , 6) , (3 , 5) , (4 , 4) , (5 , 3) , (6 , 2) and the sample space has 36 possible outcomes. Thus the probability is 5 / 36. (B) Suppose that in a bag there are 100 coins, among which exactly 1 has both sides H and all others are regular coins having one side H and the other T . Pick a random coin from the bag and ﬂip it. Assume that ﬂipping each coin shows each side with equal probabilities. Then the probability of seeing a H is 101 / 200. Ans: True.
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203_W10_Ex3-solns - EECS 203 Discrete Mathematics Winter...

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