ex3-sol09 - AMS 301.3(Fall 2009 Estie Arkin Exam 3 Solution...

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AMS 301.3 (Fall, 2009) Estie Arkin Exam 3 – Solution sketch Mean 77.9, median 80, high 100 (4 of them!), low 13. 1. (20 points) Build a generating function for a r in the following procedures. Remember to state which coe±cient solves the initial problem. You do not need to calculate the coe±cient. (a). Select r balls from an unlimited pile of white balls, and a pile of 10 red balls, such that the number of red balls must be even. Coef of x r in g ( x ) = (1 + x 2 + x 4 + ... + x 10 )(1 + x + x 2 + .... ). (b). Select 5 letters from a,b,c,d, so that b,c,d each appears at most once and if a appears then it appears exactly 4 times. (The order in which letters are selected is not important.) Coef of x 5 in g ( x ) = (1+ x 4 )(1+ x ) 3 . Common mistake: saying that a must appear exactly 4 times. Not true, it can also not appear at all. (c). A homework assignment for a calculous class is generated by the computer from a very large list of problems. The homework set will have 11 problems (the order of the problems is not important), at least 3 of the problems should be easy problems, at least 3 medium di±culty problems and at least 2 hard problems. (Note: do not distinguish between problems of the same di±culty level, i.e., you only care how many problems of each di±culty level are on the homework assignment.) Coef of x 11 in g ( x ) = ( x 3 + x 4 + ... ) 2 ( x 2 + x 3 + .... ). (d). A survey team wants to select 6 students to interview from 4 possible universities. There should be at most 3 students from Binghamton, at most 2 students from Albany, at most 2 from Bu²alo and any number of students from Stony Brook. (Note: Assume that for the purposes of
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ex3-sol09 - AMS 301.3(Fall 2009 Estie Arkin Exam 3 Solution...

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