Problem 51
Question
Find the number of ternary words that have: $$4 \leq \text { length } \leq 6$$
Step-by-Step Solution
Verified Answer
There are 1053 ternary words with lengths between 4 and 6, inclusive.
1Step 1: Finding the number of ternary words of length 4
For a ternary word of length 4, each character can have one of three possibilities. Using the counting principle, the total number of possible words is:
\[3^{4} = 3\cdot 3\cdot 3\cdot 3 = 81\]
There are 81 ternary words of length 4.
2Step 2: Finding the number of ternary words of length 5
For a ternary word of length 5, each character can still have one of three possibilities. The total number of possible words is:
\[3^{5}= 3\cdot 3\cdot 3\cdot 3\cdot 3 = 243\]
There are 243 ternary words of length 5.
3Step 3: Finding the number of ternary words of length 6
For a ternary word of length 6, each character can have one of three possibilities. The total number of possible words is:
\[3^{6}= 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3 = 729\]
There are 729 ternary words of length 6.
4Step 4: Combining the results
Now that we have the number of ternary words for lengths 4, 5, and 6, we can add these values together to find the total number of ternary words with lengths between 4 and 6, inclusive:
\[81 + 243 + 729 = 1053\]
There are 1053 ternary words with lengths between 4 and 6, inclusive.
Key Concepts
Counting PrincipleDiscrete MathematicsCombinatorics
Counting Principle
When faced with problems where you need to determine the number of possible outcomes, the counting principle is your go-to tool. This principle states that if one event can occur in 'm' ways and another independent event can occur in 'n' ways, then the total number of combined outcomes for both events is the product of 'm' and 'n'.
For example, consider a ternary word of length 4. A ternary word consists of digits 0, 1, and 2, so each character position has 3 possibilities. According to the counting principle, the number of four-character ternary words is calculated by multiplying the possibilities: \[3^4 = 3 \times 3 \times 3 \times 3 = 81\]. This calculation relies on the fact that the choice of digit for each position is independent of the others.When dealing with sequential decisions, the counting principle becomes a powerful tool to quickly compute the total number of potential outcomes without the need to list them all. Remember, the counting principle is not exclusively for digits; it applies to any scenario with multiple independent choices.
For example, consider a ternary word of length 4. A ternary word consists of digits 0, 1, and 2, so each character position has 3 possibilities. According to the counting principle, the number of four-character ternary words is calculated by multiplying the possibilities: \[3^4 = 3 \times 3 \times 3 \times 3 = 81\]. This calculation relies on the fact that the choice of digit for each position is independent of the others.When dealing with sequential decisions, the counting principle becomes a powerful tool to quickly compute the total number of potential outcomes without the need to list them all. Remember, the counting principle is not exclusively for digits; it applies to any scenario with multiple independent choices.
Discrete Mathematics
Discrete mathematics is a branch of mathematics dealing with countable, distinct elements. It includes a variety of topics such as logic, set theory, graph theory, and combinatorics. The problems in discrete mathematics often involve integers, graphs, and statements in logic, and require finite rather than continuous solutions. It does not involve real numbers with their infinite points in between.
In our exercise, we use discrete mathematics to count the number of possible ternary words of certain lengths. Since each position in the word can only take on a finite number of discrete values (0, 1, or 2), and since the words themselves have a finite length, they are well within the umbrella of discrete mathematics. Understanding the fundamentals of discrete mathematics helps to solve such problems efficiently, as you can identify and apply the correct concepts, such as the counting principle.
In our exercise, we use discrete mathematics to count the number of possible ternary words of certain lengths. Since each position in the word can only take on a finite number of discrete values (0, 1, or 2), and since the words themselves have a finite length, they are well within the umbrella of discrete mathematics. Understanding the fundamentals of discrete mathematics helps to solve such problems efficiently, as you can identify and apply the correct concepts, such as the counting principle.
Combinatorics
Combinatorics is a field of mathematics concerned with counting, arrangement, and combination of objects. It is a cornerstone of discrete mathematics and has applications in various fields including computer science, biology, and physics. Combinatorics include studying permutations, combinations, and structures such as graphs.
In the context of ternary words, combinatorics helps us understand the different arrangements of digits possible. For instance, the number of ternary words of a fixed length can be found by raising 3 (the number of possible digits) to the power of the word's length, as each position is independent and can contain any of the three digits. This concept not only simplifies the process of finding the total number of arrangements, but it also demonstrates the direct application of combinatorial principles. In more complex scenarios, combinatorics may involve more advanced techniques like generating functions or the inclusion-exclusion principle but at its core, it remains a fundamental part of understanding and solving counting problems.
In the context of ternary words, combinatorics helps us understand the different arrangements of digits possible. For instance, the number of ternary words of a fixed length can be found by raising 3 (the number of possible digits) to the power of the word's length, as each position is independent and can contain any of the three digits. This concept not only simplifies the process of finding the total number of arrangements, but it also demonstrates the direct application of combinatorial principles. In more complex scenarios, combinatorics may involve more advanced techniques like generating functions or the inclusion-exclusion principle but at its core, it remains a fundamental part of understanding and solving counting problems.
Other exercises in this chapter
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