final-s05-sol-1 - CS 70 Spring 2005 Discrete Mathematics...

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CS 70 Discrete Mathematics for CS Spring 2005 Clancy/Wagner Final Solns Problem 1. (8 points) Let a ( n ) denote the number of ternary strings of length n with no pair of adjacent 2’s. Example: “2011” is a ternary string with no pair of adjacent 2’s. (a) What are a ( 1 ) and a ( 2 ) ? Solution: a ( 1 ) = 3, a ( 2 ) = 8. Grading: 1 pt for correct answer. Solution #1: a ( n ) = 2 a ( n - 1 ) + 2 a ( n - 2 ) for n 3. JustiFcation: If the Frst digit is 0 or 1 (2 choices), then the remaining n - 1 digits can be any ternary string with no 22s ( a ( n - 1 ) choices). If the Frst digit is 2 (1 choice), then the second digit can be 0 or 1 (2 choices), and the remaining n - 2 digits can be any ternary string with no 22s ( a ( n - 2 ) choices). Solution #2: a ( n ) = 3 a ( n - 1 ) - 2 a ( n - 3 ) for n 4. JustiFcation: Take any ternary string of length n - 1 with no 22s, and append 0, 1, or 2 (3 a ( n - 1 ) choices). We need to exclude strings of the form ··· 022 and ··· 122 (2 a ( n - 3 ) choices); once we’ve eliminated these, what remains must be exactly the set of ternary strings of length n with no 22s. Common mistakes: Overcounting. ±or example, some said a ( n ) = 3 n - 3 n - 2 ( n - 1 ) , reasoning that there are 3 n - 2 ( n - 1 ) ternary strings with a 22, since there are n - 1 places where the 22 could start and 3 n - 2 options for the other n - 2 digits in the string. The problem with this reasoning is that it overcounts strings with multiple 22s; for instance, the string 22022 is counted twice, and the string 22222 is counted four times. Grading: 3 pts for a correct recurrence; 1 pt for something with a partial explanation. (b) Prove that a ( n ) 2 n + 1 for all n 2. Solution #1: Proof by strong induction. Base cases: a ( 2 ) = 8 2 2 + 1 . a ( 3 ) = 22 2 3 + 1 . Induction step: Assume a ( k ) 2 k + 1 for k = 2 , 3 ,..., n . Then for n 3, a ( n + 1 ) = 2 a ( n ) + 2 a ( n - 1 ) (by part (b)) 2 · 2 n + 1 + 2 · 2 n - 1 + 1 (by the inductive hypothesis) 2 · 2 n + 1 = 2 ( n + 1 )+ 1 . CS 70, Spring 2005, Final Solns 1
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Solution #2: For each ternary string of length n with no 22s, we can get two unique ternary strings of length n + 1 with no 22s by appending either a 0 or a 1. This shows that a ( n + 1 ) 2 a ( n ) for all n 1. Now use induction to show the desired result. Solution #3: There are 2 n ternary strings (of length n ) with no 2. There are n × 2 n - 1 more that have exactly one 2. Thus there are at least 2 n + n × 2 n - 1 ternary strings of length n with no 22. Moreover, 2 n + n × 2 n - 1 2 n + 2 × 2 n - 1 2 n + 1 for n 2. Grading: 4 pts for an impeccable proof. For induction proofs: 1 pt for all needed base cases and a correctly stated induction predicate; 3 pts for a correct and fully justi±ed inductive step. - 1 pt for backwards reason- ing, for steps with no justi±cation, for small algebraic errors, for confusing what you can assume vs. what you must prove, or for confusion between integers and booleans (e.g.,
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final-s05-sol-1 - CS 70 Spring 2005 Discrete Mathematics...

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