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Chapter 2: Problem 43

(a) If \(N\) people, including \(A\) and \(B\), are randomly arranged in a line, what is the probability that \(A\) and \(B\) are next to each other? (b) What would the probability be if the people were randomly arranged in a.circle?

Short Answer

Expert verified
The probability of $A$ and $B$ being next to each other when arranged in a line is \(\frac{2 * (N-1)!}{N!}\), and when arranged in a circle, it is \(\frac{2 * (N-2)!}{(N-1)!}\).

Step by step solution

01

Part (a) - Arrangement in a line

First, we will tackle the arrangement in a line. Let's find the total possible arrangements and the successful arrangements. Total arrangements: N people can be arranged in a line in N! ways. Now, consider A and B as a single entity. In this case, we have (N-1) entities to be arranged in a line. This can be done in (N-1)! ways. Within this arrangement, A and B can switch places, which adds another factor of 2. So successful arrangements with A and B together: 2 * (N-1)! Now, we need to find the probability: Probability = (Successful arrangements) / (Total arrangements) = \(\frac{2 * (N-1)!}{N!}\)
02

Part (b) - Arrangement in a circle

Now, let's tackle the arrangement in a circle. Again, we will find the total possible arrangements and the successful arrangements. Total arrangements: In a circle, N people can be arranged in (N-1)! ways (because circular permutations have (N-1)! different arrangements). Now, consider A and B as a single entity. In this case, we have (N-1) entities to be arranged in a circle. This can be done in (N-2)! ways. Within this arrangement, A and B can switch places, which adds another factor of 2. So successful arrangements with A and B together: 2 * (N-2)! Now, we need to find the probability: Probability = (Successful arrangements) / (Total arrangements) = \(\frac{2 * (N-2)!}{(N-1)!}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a branch of mathematics that deals with counting, arranging, and analyzing different combinations of a set of items. It is particularly useful in probability and statistics, where understanding how to count these arrangements helps to calculate probabilities.
  • Counting Principles: At the core of combinatorics are basic counting principles like the addition principle and the multiplication principle. These are used to determine the number of ways different events can occur.
  • Factorials: Factorials, denoted as \( n! \), represent the product of all positive integers up to \( n \). Factorials are pivotal in combinatorics for calculating permutations and combinations, outlining how many ways objects can be ordered or selected.
  • Applications: Combinatorial methods apply to various problems, such as arranging people in a line or analyzing networks. In our context, it helps count possible arrangements of individuals in line or circle formats.
The combination of these concepts provides a robust framework for solving complex arrangement and counting problems efficiently.
Permutations
Permutations are an essential concept in combinatorics, especially when considering the order of arrangements. A permutation refers to an arrangement or sequence of items where the order matters.
  • Linear Arrangements: For \( N \) distinct items, there are \( N! \) possible permutations, because every item can be placed in any position, sequentially turning previous positions unavailable for the next items.
  • Circular Arrangements: When arrangements occur in a circle, the formula changes slightly to \( (N-1)! \), because circular placements allow for rotational symmetry; the start position is not fixed, reducing the total ways.
  • Permutation Applications: Understanding permutations is vital for solving problems where the order of arrangement is key, such as determining the sequence of objects or the arrangement of people in a line or circle.
Mastering permutations provides a foundational tool for further investigating and solving complex probability questions.
Probability in Line Arrangements
When determining the probability of an event, like two people standing next to each other in a line, we use the principles of linear permutations mixed with the concept of favorable outcomes over possible outcomes.
  • Total Arrangements: When people are arranged in a line, all possibilities are captured by \( N! \), where \( N \) is the number of individuals.
  • Favorable Arrangements: Consider two specific people as a single unit or 'block.' This 'block' can interchange positions within itself contributing to additional arrangements, resulting in \( 2 \times (N-1)! \) successful sequences.
  • Probability Calculation: The probability is given as a ratio of successful sequences to total sequences: \( \frac{2 \times (N-1)!}{N!} \). This simplifies to help find how likely two people will be side by side when lined up.
Breaking arrangements into simple components helps visualize and calculate probabilities efficiently, providing clear solutions to arrangement problems.
Probability in Circular Arrangements
Circular arrangements bring a twist to the probability calculation because of their inherently symmetrical nature. Calculating probabilities in a circle requires a different approach than line arrangements.
  • Total Arrangements: In circular setups, \( N \) people are arranged in \( (N-1)! \) ways because the arrangement is considered the same when rotated.
  • Favorable Arrangements: By grouping two individuals as a single block and considering their mutual arrangement within the block, successful circular permutations become \( 2 \times (N-2)! \).
  • Probability Determination: The formula used: \( \frac{2 \times (N-2)!}{(N-1)!} \), encapsulates the nuanced placements of such distinct groupings in a circle, focusing on how likely two specific individuals are next to each other.
Such calculations help gauge how group dynamics change when shifted from linear to circular formats, offering insight into rotationally symmetric scenarios.

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Most popular questions from this chapter

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