15b_01f_fe

15b_01f_fe - Math 15B Final Exam 6 December 2001 Please put...

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Unformatted text preview: Math 15B Final Exam 6 December 2001 Please put your name and ID number on your blue book. The exam is CLOSED BOOK, but TWO PAGES OF NOTES ARE ALLOWED. Calculators are NOT ALLOWED. You need not evaluate binomial coefficients. You must show your work to receive credit. 1. In each case, give an example or explain why none exists . (a) A permutation f of { 1 , 2 , 3 , 4 , 5 } such that, for some x { 1 , 2 , 3 , 4 , 5 } , f 20 ( x ) 6 = x . (b) A permutation f of { 1 , 2 , 3 , 4 , 5 } such that, for every x { 1 , 2 , 3 , 4 , 5 } , f 20 ( x ) 6 = x . (c) A tree with exactly 10 vertices and exactly 10 edges. 2. In each case, give an example or explain why none exists . (a) A function f ( n ) such that f ( n ) is O ( n 2 ) but f ( n ) is not ( n 2 ). (b) A function f ( n ) such that f ( n ) is O ( n log n ) but f ( n ) is not O ( n 2 ). (c) A probability space ( U, P ) and two subsets S and T of U such that P ( S ) = P ( T ) = 2 / 3 and S 6 = T ....
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15b_01f_fe - Math 15B Final Exam 6 December 2001 Please put...

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