qswd - MATH 239 Quiz 2 No calculators or other aids may be...

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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 a span of 10 days. How many ways can he do this if he plays at least one, but no more than nine games every day? 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 denote the number of games that Eddie plays on days 1 through 10 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 = n. Therefore, we need to count the number of 10-tuples whose sum is equal to n . The condition that he plays at least one, but no more than nine games every day means that 1 a i 9 for each i . The set S of all 10-tuples a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , a 9 , a 10 satisfying this condition is given by S = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } 10 .
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