Assn3 - 5 Before the midterm test each of 30 students...

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1. Find the generating functions for the following sequences: 0 , 1 , 4 , 13 , . . . ( a n = 3 n - 1 2 ) 7 , 10 , 15 , 22 , . . . ( a n = n 2 + 2 n + 7) 3 , 3 , 3 , 3 , 4 , 5 , 6 , 7 , . . . ( a n = n for n 3, a 0 = a 1 = a 2 = 3) 2 , 1 , 5 , 7 , 17 , . . . ( a n = 2 n + 1 for n even, a n = 2 n - 1 for n odd) 2. Find the generating function for the number of partitions of n to dis- tinct even summands. 3. Find the partial fractions decomposition of f ( x ) = 1 (1 - x ) 3 (1+ x ) , and use it to ±nd a formula for the coe²cient of x n in the power series expansion of f ( x ). 4. Find the generating function for the number of solutions to x 1 + x 2 + x 3 = n , where x 1 , x 2 and x 3 are nonnegative integers, x 1 10, x 2 100 and x 3 is even. Use it together with the result of the third exercise to ±nd a formula for the number of solutions.
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Unformatted text preview: 5. Before the midterm test, each of 30 students prepares a cheatsheet and hides it in his desk ( don’t do that!!! ). On the day of the test, the teacher seats the students randomly. As they did not prepare otherwise and they are unable to decipher other’s cheatsheets, those that end up sitting at their own desk will pass the exam, while the rest of the class will fail. What is the probability that • everyone will pass? • everyone will fail? • exactly 7 of the students will pass?...
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