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ex3-sol07 - AMS 301.3(Fall 2007 Estie Arkin Exam 3 Solution...

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AMS 301.3 (Fall, 2007) Estie Arkin Exam 3 – Solution sketch Mean 82.32, median 85, high 100 (2 of them!), low 43. 1. (20 points) Build a generating function for the the number of election outcomes in the race for class president if there are 4 candidates and 30 students votes. (You do not need to calculate the coefficient.): Each part is independent of the others! (a). Every candidate gets at least one vote? Let e i be the number of votes for candidate i , so e 1 + e 2 + e 3 + e 4 = 30, e i 1, gives the generating function g ( x ) = ( x + x 2 + x 3 + · · · ) 4 , we want the coefficient of x 30 . (b). Tiffany (one of the candidates) gets at least 4 votes? g ( x ) = (1+ x + x 2 + x 3 + · · · ) 3 ( x 4 + x 5 + · · · ), we want the coefficient of x 30 . (c). Voters are allowed to abstain (vote for none of the 4 candidates)? Let e 5 be the number of voters that abstain, we have e 1 + e 2 + e 3 + e 4 + e 5 = 30, e i 0 and so the generating function is g ( x ) = (1 + x + x 2 + x 3 + · · · ) 5 , we want the coefficient of x 30 .
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