Problem 78 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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.

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Key Concepts

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’s 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—the 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—just like we fixed one object in the circular permutation to simplify the count.