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Chapter 7: Problem 78

Suppose \(n\) distinct objects are arranged in a circle. Show that the number of (different) circular arrangements of the \(n\) objects is \((n-1) !\) Hint: Consider the arrangement of the five letters \(A, B, C, D\), and \(E\) in the accompanying figure. The permutations \(A B C D E, B C D E A\), \(C D E A B, D E A B C\), and \(E A B C D\) are not distinguishable. Generalize this observation to the case of \(n\) objects.

Short Answer

Expert verified
In summary, for $n$ distinct objects arranged in a circular fashion, there are \((n-1)!\) different circular arrangements, since we fix one object as a reference point and arrange the remaining \((n-1)\) objects around it, taking into account that rotations are not considered distinct permutations.

Step by step solution

01

Understand the Example

First, let's have a closer look at the example given in the hint. Here, we have 5 letters, A, B, C, D, and E, arranged in a circle. There are 5! ways to arrange these letters in a linear fashion. However, since this is a circular arrangement, we need to take into consideration that certain linear permutations will appear identical when arranged around a circle. For example, ABCDE, BCDEA, CDEAB, DEABC, and EABCD are all considered the same when arranged in a circle because they are just rotations of the same arrangement.
02

Identify the Key Concept

To generalize the example, we must identify the key concept that distinguishes circular arrangements from linear arrangements. The crucial point here is the concept of indistinguishability amongst different rotations of the same arrangement. A circular arrangement is a unique set of objects ordered around a circle where we don't consider rotations as distinct permutations.
03

Generalize the Problem

Now, let's generalize the problem for n distinct objects placed in a circle. In this case, we can think of fixing one object and arranging the remaining (n-1) objects around it. Suppose we fix a particular object A in a distinct position (say at the top). We now have the task of arranging the remaining (n-1) objects around A in a circular order. This can be done in the same way as arranging them in a linear order, but since we have fixed A as our reference point, we don't need to worry about taking rotations of our arrangement into account anymore.
04

Compute the Number of Arrangements

Since there are (n-1) objects left to arrange, there are (n-1)! ways to do it, taking into account the fact that our reference point A is fixed, and hence no rotations of the arrangement should be considered. Therefore, the number of distinct circular arrangements for n objects is given by (n-1)!. We have concluded that for n distinct objects arranged in a circle, there are (n-1)! different circular arrangements.

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Key Concepts

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

Permutations
Permutations are the different ways in which a set of objects can be ordered or arranged. The concept of permutations is vital in combinatorics, a branch of mathematics concerned with counting, arrangement, and decision-making.

Understanding permutations is essential when dealing with various ordering problems, whether it's arranging books on a shelf, selecting a batting order for a cricket team, or encrypting data with a complex algorithm.

Let鈥檚 take a simple example of 3 objects: A, B, and C. The different ways (permutations) of arranging these objects are: ABC, ACB, BAC, BCA, CAB, and CBA. Thus, for 3 objects, there are 6 permutations. In general, if we have n distinct objects, the number of linear permutations is given by the factorial of n, denoted as n!.
Factorial Notation
Factorial notation is a mathematical expression that represents the product of all positive integers from 1 to that number. It's symbolized by an exclamation point (!) right after the number. For instance, the factorial of 4, written as 4!, is calculated as 4 x 3 x 2 x 1, which equals 24.

Factorials grow exponentially fast with the increase of the number, which is why they appear so frequently in combinatorial problems involving large sets. The factorial function is also fundamental when calculating permutations, as it denotes the total number of ways you can arrange a set of objects.

Understanding Factorial Zero

One peculiar aspect of factorial notation is that 0! is defined to be 1. This definition is consistent within the realm of combinatorics since there is exactly one way to arrange zero objects鈥攖he arrangement that has nothing in it!
Arrangements in Combinatorics
Combinatorics is the field of mathematics that studies collections of objects that satisfy specified criteria. In particular, it looks at the 'counting' of objects and their arrangements within certain rules. When it comes to circular permutations, the approach varies slightly from linear arrangements because the starting point in a circle is arbitrary.

To elaborate, imagine a necklace with different colored beads. Due to its circular nature, rotating the necklace does not change the order of beads, hence it is considered the same arrangement. This is where (n-1)! becomes important. By fixing one element and arranging the remaining, we have a simple way to count arrangements without redundancies caused by rotations.

Decomposing Complex Arrangements

Advanced combinatoric problems often require decomposing a complex arrangement into simpler parts. By breaking down a problem, such as seating arrangements at a round table, we can create a step-by-step process to find the solution effectively鈥攋ust like we fixed one object in the circular permutation to simplify the count.

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Most popular questions from this chapter

In a poll conducted among 2000 college freshmen to ascertain the political views of college students. the accompanying data were obtained. Determine the empirical probability distribution associated with these data. $$ \begin{array}{lccccc} \hline \text { Political Views } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Respondents } & 52 & 398 & 1140 & 386 & 24 \\ \hline \end{array} $$ A: Far left B: Liberal C: Middle of the road D: Conservative E: Far right

A study conducted by the Corrections Department of a certain state revealed that 163,605 people out of a total adult population of \(1,778,314\) were under correctional supervision (on probation, on parole, or in jail). What is the probability that a person selected at random from the adult population in that state is under correctional supervision?

According to a study of 100 drivers in metropolitan Washington, D.C., whose cars were equipped with cameras with sensors, the distractions and the number of incidents (crashes, near crashes, and situations that require an evasive maneuver after the driver was distracted) caused by these distractions are as follows: $$ \begin{array}{lccccccccc} \hline \text { Distraction } & A & B & C & D & E & F & G & H & I \\ \hline \text { Driving Incidents } & 668 & 378 & 194 & 163 & 133 & 134 & 111 & 111 & 89 \\ \hline \end{array} $$ where \(A=\) Wireless device (cell phone, PDA) \(\begin{aligned} B &=\text { Passenger } \\ C &=\text { Something inside car } \\ D &=\text { Vehicle } \\ E &=\text { Personal hygiene } \\ F &=\text { Eating } \\ G &=\text { Something outside car } \\ H &=\text { Talking/singing } \\ I &=\text { Other } \end{aligned}\) If an incident caused by a distraction is picked at random, what is the probability that it was caused by a. The use of a wireless device? b. Something other than personal hygiene or eating?

Eight players, \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}, \mathrm{G}\), and \(\mathrm{H}\), are com- peting in a series of elimination matches of a tennis tournament in which the winner of each preliminary match will advance to the semifinals and the winners of the semifinals will advance to the finals. An outline of the scheduled matches follows. Describe a sample space listing the possible participants in the finals.

What is the probability that a roulette ball will come to rest on an even number other than 0 or 00 ? (Assume that there are 38 equally likely outcomes consisting of the numbers \(1-36,0\), and 00 .)
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