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Unformatted text preview: MATH 239 Quiz 2 No calculators or other aids may be used. Show all your work. 1. Eddie plans to play n games of chess over the next 11 days. How many ways can he do this if the number of games that he plays each day is between 3 and 8? Express your answer as the coefficient of a rational function (i.e. P ( x ) Q ( x ) where P ( x ) ,Q ( x ) are polynomials). You do not need to compute the coefficient explicitly. Solution. Let a 1 ,a 2 ,a 3 ,a 4 ,a 5 ,a 6 ,a 7 ,a 8 ,a 9 ,a 10 ,a 11 denote the number of games that Eddie plays on days 1 through 11 respectively. Then the number of ways to play n games is equal to the number of solutions to the equation a 1 + a 2 + a 3 + a 4 + a 5 + a 6 + a 7 + a 8 + a 9 + a 10 + a 11 = n. Therefore, we need to count the number of 11-tuples whose sum is equal to n . The condition that he plays at least three, but no more than eight games every day means that 3 a i 8 for each i . The set S of all 11-tuples a 1 ,a 2 ,a 3 ,a 4 ,a 5 ,a 6 ,a 7 ,a 8 ,a 9 ,a...
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