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hw08 - How many di²erent answer keys are possible 1(c A...

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In the Name of God Fall 2005 Discrete Mathematics Homework 8 Due: Azar 12 Problem 1. In this problem, we’ll use generating functions to solve the recurrence: t 0 = 0 , t 1 = 1 , t n = 5 t n - 1 - 6 t n - 2 (a) Find a closed-form generating function F ( x ) for the sequence ( t 0 , t 1 , t 2 , ··· ) . (b) Rewrite this generating function as a sum of fractions of the form c 1 - rx where c and r are constants. (c) Expand each fraction using the fact: 1 1 - rx = 1 + rx + r 2 x 2 + r 3 x 3 + ··· Combine these expansions to obtain a closed-form expression for t n . Problem 2. Use generating functions to ±nd an explicit formula for the Fibonacci numbers. Problem 3. For each of the following generating functions, provide a closed formula for the sequence it determines. (a) (1+ x 3 ) (1+ x ) 3 (b) x (1+ x + x 2 ) Problem 4. (a) A true-false exam has 15 questions, exactly three answers are false, but no two falses are ever consecutive. How many di²erent answer keys are possible? (b) A true-false exam has 20 questions, but no two falses are ever consecutive.

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Unformatted text preview: How many di²erent answer keys are possible? 1 (c) A multiple-choice exam has 50 questions, and each question ofers Four choices: A, B, C, and D. Each question has exactly one answer, ten are B, and another ten are D. How many diferent answer keys to the exam satisFy these constraints? Problem 5. ±ind a closed-Form generating Function For the sequence (1 , 3 , 5 , 7 , 9 , 11 , ··· ) Problem 6. How many ways are there to tile an n × 3 board using 2 × 1 dominoes? Model this counting problem using a coupled set oF recurrence relations which in theory could be solved to give you a closed Formula. Problem 7. How many ways are there to build a 2 × 2 × n pillar using 1 × 1 × 2 bricks? Model this counting problem using a coupled set oF recurrence relations. 2...
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• Spring '11
• Tn, Recurrence relation, Fibonacci number, Generating function, diﬀerent answer keys, closed-form generating function

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hw08 - How many di²erent answer keys are possible 1(c A...

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