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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 02/06/2011 for the course ECE 423 taught by Professor Dolatabadi during the Spring '11 term at Amirkabir University of Technology.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online