hw4-solns - Illinois Institute of Technology Department of...

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Unformatted text preview: Illinois Institute of Technology Department of Computer Science Solutions to Homework Assignment 3 CS 330 Discrete Structures Spring Semester, 2008 1. Solve using recurrences and operator method. (a) Page 458, Problem 28. i. Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one stair, or two, or three stairs at a time. Let a n be the number of ways to climb n stairs. We can start by taking 1 stair then n- 1 stairs in a n- 1 ways, or we can start with 2 stairs then n- 1 stairs in a n- 1 ways, or we can start with 3 stairs, then climb n- 3 stairs in a n- 3 ways. This gives us a recurrence relation. a n = a n- 1 + a n- 2 + a n- 3 for n 3 ii. What are the initial conditions? a = 1, a 1 = 1, a 2 = 2 iii. Operator method a n +3- a n +2- a n +1- a n = 0 ( E 3- E 2- E- 1) = 0 ( E- 1 )( E- 2 )( E- 3 ) a n = 1 n + 2 n + 3 n . iv. Solve for constants using initial conditions. a = 1 a 1 = 1 a 2 = 2 solve with cubic equation. v. How many ways to climb a flight of eight stairs? a n = a n- 1 + a n- 2 + a n- 3 a 3 = 2 + 1 + 1 = 4 a 4 = 4 + 2 + 1 = 7 a 5 = 7 + 4 + 2 = 13 a 6 = 13 + 7 + 4 = 24 a 7 = 24 + 13 + 7 = 44 a 8 = 44 + 24 + 13 = 81 CS 330Spring, 2008 2 Solutions to Homework Assignment 3 (b) Page 458, Problem 30....
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This homework help was uploaded on 04/15/2008 for the course CS 330 taught by Professor Reingold,edwardm. during the Spring '08 term at Illinois Tech.

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hw4-solns - Illinois Institute of Technology Department of...

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