AP 09 - Feb 20

AP 09 - Feb 20 - S = { 1 , 2 , 3 , . . . , 12 } contain no...

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1 Math 2602 M1/M2 Spring 2007 Additional Problems 9 Generating Functions 1 Tuesday, Feb 20th 1. Find the coefficient of x 10 in (2 - 3 x ) - 18 . 2. Determine the coefficient of x 20 in ( x 3 + x 4 + x 5 + · · · ) 5 . 3. Determine the coefficient of x 10 in 1 ( x - 3)( x - 2) 2 . Hint: use the partial fraction decomposition to see that 1 ( x - 3)( x - 2) 2 = 1 ( x - 3) - 1 x - 2 - 1 ( x - 2) 2 . 4. Determine the coefficient of x 15 in x 3 - 5 x (1 - x ) 3 . 5. Determine the coefficient of x 15 in (1 + x ) 4 (1 - x ) 4 . 6. In how many ways can a police captain distribute 30 rifle shells to five police officers so that each officer gets at least three shells, but not more than eight. 7. Use generating functions to determine how many five-element subsets of
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Unformatted text preview: S = { 1 , 2 , 3 , . . . , 12 } contain no consecutive integers. 8. In how many ways can two dozen identical robots be assigned to four assembly lines with (a) at least three robots assigned to each line ? (b) at least three, but no more than nine, robots assigned to each line ? 9. Show that (1-4 x )-1 / 2 generates the sequence 2 n n . 10. If a fair die is rolled 12 times, in how many ways is the sum of the roll 35? 1 T. Yolcu, School of Mathematics at Georgia Tech. Email: tyolcu@math.gatech.edu ....
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