Chapter 7: Problem 17
How many ways are there to seat eight people at a round table? How many ways if Smith and Jones cannot be seated next to each other?
Short Answer
Expert verified
Total ways: 5040. Non-adjacent Smith and Jones: 2520 ways.
Step by step solution
01
Understand Circular Permutations
To find the number of ways to arrange eight people at a round table, we use the concept of circular permutations. The formula for circular permutations of \( n \) objects is \((n-1)!\).
02
Calculate Total Arrangements
Using the formula from Step 1, calculate the total number of arrangements for 8 people: \((8-1)! = 7!\).
03
Calculate \(7!\)
Calculate \(7!\): \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\). There are 5040 ways to seat eight people at a round table.
04
Determine Arrangements for Smith and Jones
If Smith and Jones cannot sit next to each other, first find the number of ways they can sit together. Treat Smith and Jones as a single 'block'. This block can be seated in \( (8-1)! = 7! \) different ways, but since they can swap places within their block, multiply by 2. So, they can sit together in \( 2 \times 7! = 2 \times 5040 = 10080 \) ways.
05
Calculate Valid Arrangements for No Adjacent Smith and Jones
Subtract the arrangements where Smith and Jones sit together from the total arrangements: \(5040 - 10080/2 = 2520\). Thus, there are 2520 valid arrangements where Smith and Jones do not sit next to each other.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Round Table Arrangements
Arranging people around a round table presents an interesting problem in mathematics and combinatorics. Unlike linear arrangements, where the sequence is straightforward, circular arrangements have no fixed starting point. This means a single circular arrangement can appear in different forms by rotating the people around the table.
In circular permutations, to count distinct arrangements, we fix one person and arrange the rest. This reduces the problem to arranging the remaining people in a line, effectively making the formula for the number of arrangements \( (n-1)! \), where \( n \) is the number of people.
Therefore, for eight people around a round table, the total number of ways they can be seated is calculated by \( 7! \). This simplifies the complexity, helping us avoid overcounting that typically occurs when arranging linearly.
In circular permutations, to count distinct arrangements, we fix one person and arrange the rest. This reduces the problem to arranging the remaining people in a line, effectively making the formula for the number of arrangements \( (n-1)! \), where \( n \) is the number of people.
Therefore, for eight people around a round table, the total number of ways they can be seated is calculated by \( 7! \). This simplifies the complexity, helping us avoid overcounting that typically occurs when arranging linearly.
Factorial Calculation
Factorials are a fundamental concept in permutation and combination problems. A factorial of a number is the product of all positive integers less than or equal to that number. Denoted as \( n! \), it signifies the total number of ways to arrange \( n \) distinct objects linearly.
For instance, calculating \( 7! \), involves multiplying every integer from 1 to 7 together: \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \). This helps us understand the vast number of possible permutations even with relatively small numbers.
In our round table example, \( 7! \) gives the number of ways to arrange 8 people circularly, illustrating the efficiency of using factorials in permutation calculations. Learning to compute factorials is not only crucial for theoretical exercises but also for solving real-world problems involving arrangements.
For instance, calculating \( 7! \), involves multiplying every integer from 1 to 7 together: \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \). This helps us understand the vast number of possible permutations even with relatively small numbers.
In our round table example, \( 7! \) gives the number of ways to arrange 8 people circularly, illustrating the efficiency of using factorials in permutation calculations. Learning to compute factorials is not only crucial for theoretical exercises but also for solving real-world problems involving arrangements.
Combinatorics
Combinatorics is the mathematical study of counting, specifically, how to arrange different sets of objects. It covers various topics, including permutations and combinations, which are integral to understanding problems like round table arrangements.
Permutations refer to the different ways of arranging objects, and with constraints, it requires special strategies. For our example with eight people, combinatorics helps us count these arrangements efficiently by recognizing the unique nature of circular tables.
Combinatorics often involves recognizing patterns and using formulas like \( (n-1)! \) for circular permutations to simplify calculations, enabling us to deal with more complex constraints easily.
This field not only deals with abstract theoretical problems but is also applied in fields like computer science, cryptography, and logistics to solve real-world challenges involving complex arrangements.
Permutations refer to the different ways of arranging objects, and with constraints, it requires special strategies. For our example with eight people, combinatorics helps us count these arrangements efficiently by recognizing the unique nature of circular tables.
Combinatorics often involves recognizing patterns and using formulas like \( (n-1)! \) for circular permutations to simplify calculations, enabling us to deal with more complex constraints easily.
This field not only deals with abstract theoretical problems but is also applied in fields like computer science, cryptography, and logistics to solve real-world challenges involving complex arrangements.
Permutation Constraints
When arranging objects, constraints can significantly affect the number of possible permutations. For instance, placing a condition like "Smith and Jones cannot sit next to each other" in a seating arrangement modifies the calculation process.
To handle constraints, we often use the method of complementary counting or consider the constraint as a block. With the given example, treating Smith and Jones as one unit reduces our arrangement problem to simpler terms.
First, calculate the arrangements where they sit together, considered as a single entity. Multiply by the number of ways they can swap places within the block (in this case, \( 2! = 2 \)). Subtract these from the total permutations to find valid configurations without the constraint.
This approach helps simplify complex constraints and neatly ties into the main principles of combinatorics, allowing us to consider unique conditions within common permutation problems.
To handle constraints, we often use the method of complementary counting or consider the constraint as a block. With the given example, treating Smith and Jones as one unit reduces our arrangement problem to simpler terms.
First, calculate the arrangements where they sit together, considered as a single entity. Multiply by the number of ways they can swap places within the block (in this case, \( 2! = 2 \)). Subtract these from the total permutations to find valid configurations without the constraint.
This approach helps simplify complex constraints and neatly ties into the main principles of combinatorics, allowing us to consider unique conditions within common permutation problems.
