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184a_03f_fe - Math 184A Final Exam Please put your name and...

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Math 184A Final Exam Fall 2003 Please put your name and ID number on your blue book. CLOSED BOOK, but you may have TWO PAGES of notes. Calculators are NOT allowed. You need not simplify the arithmetic in your answers. You must show your work to receive credit. 1. (8 pts.) In each case, give an example or explain why none exists . (a) A permutation f of { 1 , 2 , 3 , 4 , 5 } such that f 20 is not the indentity permutation. (The identity permutation is the function g such that g ( x ) = x for all x in the domain.) (b) A simple graph with 5 vertices and 12 edges. 2. (16 pts.) Committees are to be formed from 5 Democrats and 8 Republicans. A committee must be as balanced as possible; that is, the number of Republicans and Democrats on the committee must be as equal as possible. (a) How many ways can a 7 member committee be formed? (b) How many ways can both a 7 member committee and a 5 member committee be formed if no person can be on both committees?
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