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hw4-solns

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

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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 0 = 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 - φ 1 )( E - φ 2 )( E - φ 3 ) a n = αφ 1 n + βφ 2 n + γφ 3 n . iv. Solve for constants using initial conditions. a 0 = 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

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CS 330—Spring, 2008 2 Solutions to Homework Assignment 3 (b) Page 458, Problem 30. i. Find a recurrence relation for the number of ternary strings (e.g. 0 , 1 , 2) that contain two consecutive 0’s.
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hw4-solns - Illinois Institute of Technology Department of...

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