184a_01w_1e

# 184a_01w_1e - “no”or “don’t know” The article...

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Math 184A First Exam 31 January 2001 Please put your name and ID number on your exam. If you are not using a blue book, put your name on every page. The exam is CLOSED BOOK. Calculators are allowed. You must show your work to receive credit. 1. For each of the following, EITHER give an example of the thing that is described OR explain why none exists. (a) A surjection from { 1 , 2 , 3 } to { a, b, c, d } . (b) An injection from { 1 , 2 , 3 } to { a, b, c, d } . (c) A permutation f of { 1 , 2 , 3 , 4 } such that f 12 is NOT the identity function. The identity function is the function g such that g ( x )= x for all x . Remember that f 12 ( x )is f ( f ( ··· f ( x ))), not ( f ( x )) 12 . (d) An involution f of { 1 , 2 , 3 , 4 , 5 , 6 } with exactly 4 cycles. 2. How many 6 card hands contain 3 pairs? Important : Give a careful justiﬁcation of how you found your answer, not just a bunch of numbers multiplied together without explanation. 3. A magazine article lists 8 yes/no questions. Each question must be answered “yes,”
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Unformatted text preview: “no”or “don’t know”. The article says that a person is credulous if he or she answers at least 6 of the questions with a “yes”. How many ways are there to answer the questions so that you receive the label “credulous?” 4. Let S be an n-set. It was shown in the text (and in class) that the number of subsets that can be formed from S is 2 n . Generalize this: State and prove a formula for the number of multisets that can be formed from S where each element is repeated at most k times . When k = 1, your formula should become 2 n . 5. Let S ( n, k ) be the Stirling numbers of the second kind; that is, S ( n, k ) is the number of ways to partition an n-set into k unordered blocks. Prove that S ( n, k ) = n X j =1 ± n-1 j-1 ¶ S ( n-j, k-1) for n > 0 and k > 0. END OF EXAM...
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