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Chapter 1: Problem 24

Given \(n\) distinct objects, determine in how many ways \(r\) of these objects can be arranged in a circle, where arrangements are considered the same if one can be obtained from the other by rotation.

Short Answer

Expert verified
The number of arrangements of \(r\) objects from \(n\) distinct ones in a circular arrangement is given by the formula \(\frac{(n-1)!}{(n-r)!}\).

Step by step solution

01

Understand Problem Statement

The task asks to find the number of arrangements of \(r\) objects out of \(n\) distinct objects, when arranged in a circle. Here, rearrangements by rotation are considered the same. Thus, we are dealing with a problem of circular permutations.
02

Apply Circular Permutation Formula

The formula for circular permutations of \(r\) objects from \(n\) distinct objects is \((n-1)P(r-1)\) or equivalently \(\frac{(n-1)!}{(n-r)!}\). This formula takes into account that the circular arrangement has rotational symmetry.
03

Generalize the Solution with Variables

In order to give a generalized solution to the problem, it remains to write down the formula for circular permutations using the variables \(n\) (number of distinct objects) and \(r\) (number of chosen arrangements) as follows: The number of circular permutations = \(\frac{(n-1)!}{(n-r)!}\)

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