184a_05w_1e

184a_05w_1e - form a 4-person committee that contains at...

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Math 184A First Exam (40 points) 26 January 2005 Please put your name and ID number on your blue book. The exam is CLOSED BOOK except for one page of notes. Calculators are NOT allowed. You must show your work to receive credit. 1. (8 pts.) Prove that the number of ordered lists without repeats (including the empty list) that can be constructed from an n -set is nearly n ! e . Hint : By Taylor’s theorem, e is nearly 1 + 1 / 1! + 1 / 2! + 1 / 3! + · · · + 1 /n !. 2. (8 pts.) How many 6-card hands contain 3 pairs? As usual: We are assuming a deck of 52 cards with 13 face values and 4 suits. A pair is two cards with the same face values. Three pairs means each pair has a di±erent face value from the other two. You may leave factorials and binomial coe²cients in your answer. 3. (8 pts.) The departments of Engineering, Mathematics and Computer Science at Small College have 5 (³ve), 7 (seven) and 6 (six) members respectively. They want to
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Unformatted text preview: form a 4-person committee that contains at least one person from each department. How many possible committees are there? You may leave factorials and binomial coecients in your answer. 4. (8 pts.) Describe all permutations f of { 1 , 2 , 3 , 4 , 5 } such that f k is not the identity for any positive k { 1 , 2 , 3 , 4 , 5 } . You must justify your answer. Hint : Look at cycle lengths. 5. (8 pts.) A k-part partition of n is a k-multiset of positive integers whose sum is n . For example the 2-part partitions of 6 are { 1 , 5 } , { 2 , 4 } and { 3 , 3 } . (a) Prove that there are exactly m 2-part partitions of 2 m when m > 0. (b) State and prove a formula for the number of 2-part partitions of 2 m + 1 when m > 0. Hint : If you do not see the formula right away, list the partitions for m = 1, m = 2 and maybe m = 3. END OF EXAM...
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This note was uploaded on 06/18/2009 for the course MATH 184a taught by Professor Staff during the Winter '08 term at UCSD.

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