Chapter 13.2, Problem 19PPS

(a)

To determine

To find:the number of ways can the points on circle be named using letter $A$ through $E$ .

Answer to Problem 19PPS

There is a $720$ ways.

Explanation of Solution

Given:

Seven point are chosen from the circle.

  

Concept used:

The number of ways of arranging $n$ item in order $=n!$ .

The probability formula is used to compute the probability of an event to occur as:

  $P\left(A\right)=\frac{\text{Number of favorable outcome}}{\text{Total Number of favorable outcomes}}$ .

Calculation:

The number of distinguishable permutations of n objects arranged in a circle with no fixed reference point is:

  $\frac{n!}{n}=\left(n-1\right)!$ .

The  $7$  points can be named in a circular way in  $6!=720$ different ways.

Hence, there is a $720$ ways.

(b)

To determine

To find:the number of possible arrangements if one point on circle is fixed.

Answer to Problem 19PPS

There is $5040$ ways.

Explanation of Solution

Given:

Seven point are chosen from the circle.

  

Concept used:

The number of ways of arranging $n$ item in order $=n!$ .

The probability formula is used to compute the probability of an event to occur as:

  $P\left(A\right)=\frac{\text{Number of favorable outcome}}{\text{Total Number of favorable outcomes}}$ .

Calculation:

Since the points are named with a fixed reference points, this is a linear permutation. There are $7!$ or $5040$ ways in which the points can be named.

Hence, there is $5040$ ways.