Short Answer

How many ternary strings of length six do not contain two consecutive $0' s$ or two consecutive $1' s$ ?

There are 239 ternary strings of length six that do not contain two consecutive $0' s$ or two consecutive $1' s$ .

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Step by Step Solution

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TABLE OF CONTENTS

Step 1: Given data

A string that contains only $0' s$ , $1' s$ and $2' s$ is called a ternary string.

Step 2: Concept used of recurrence relation

A recurrence relation is an equation that recursively defines a sequence where the next term is a function of the previous terms (Expressing $F_{n}$ as some combination of $F_{i}$ with $i < n)$ .

Step 3: Find the number of ternary strings

We will compute $a_{2}$ through $a_{6}$ using the recurrence relation:

$a_{2} = a_{1} + 2 a_{0} + 2 = 3 + 2 \cdot 1 + 2 = 7 a_{3} = a_{2} + 2 a_{1} + 2 a_{0} + 2 = 7 + 2 \cdot 3 + 2 \cdot 1 + 2 = 17 a_{4} = a_{3} + 2 a_{2} + 2 a_{1} + 2 a_{0} + 2 = 17 + 2 \cdot 7 + 2 \cdot 3 + 2 \cdot 1 + 2 = 41 a_{5} = a_{4} + 2 a_{3} + 2 a_{2} + 2 a_{1} + 2 a_{0} + 2 = 41 + 2 \cdot 17 + 2 \cdot 7 + 2 \cdot 3 + 2 \cdot 1 + 2 = 99 a_{6} = a_{5} + 2 a_{4} + 2 a_{3} + 2 a_{2} + 2 a_{1} + 2 a_{0} + 2 = 99 + 2 \cdot 41 + 2 \cdot 17 + 2 \cdot 7 + 2 \cdot 3 + 2 \cdot 1 + 2 = 239$

Thus there are 239 ternary strings of length 6 that do not contain two consecutive $0' s$ or two consecutive $1' s$ .

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Discrete Mathematics and its Applications

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Short Answer

How many ternary strings of length six do not contain two consecutive $0' s$ or two consecutive $1' s$ ?

Step by Step Solution

Step 1: Given data

Step 2: Concept used of recurrence relation

Step 3: Find the number of ternary strings

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Discrete Mathematics and its Applications

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Short Answer

How many ternary strings of length six do not contain two consecutive 0'sor two consecutive 1's ?

Step by Step Solution

Step 1: Given data

Step 2: Concept used of recurrence relation

Step 3: Find the number of ternary strings

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How many ternary strings of length six do not contain two consecutive $0' s$ or two consecutive $1' s$ ?