Chapter 13.2, Problem 20PPS

(a)

To determine

To find:The number of ways that the seats on the carousel be randomly filled with $9$ people.

Answer to Problem 20PPS

The number of ways is $362,880$ .

Explanation of Solution

Given:

A carousel has $7$ horses.

  

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 ways in which the $9$ seats can be filled is the permutation of $9$ people taken $9$ at a time. So, the number of ways is:

  $9!=362,880$ .

Hence, the number of ways is $362,880$ .

(b)

To determine

To find:the probability that friends will end up in the bench seat if carousel is filled randomly.

Answer to Problem 20PPS

The probability is $\frac{1}{36}$ .

Explanation of Solution

Given:

A carousel has $7$ horses.

  

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 total number of possible outcomes is $362,880$ . Since, the carousel is filled with a fixed reference point, this is a linear permutation. So, there are $7!$ or $5040$ ways to arrange the remaining $7$ people after you and your friend are placed in the bench seat. Also, for each arrangement there is an alternate arrangement by interchanging the position of you and your friend. Therefore, the probability is:

  $\frac{2\left(5040\right)}{3262880}=\frac{1}{36}$ .

Hence, the probability is $\frac{1}{36}$ .

(c)

To determine

To find:the probability that a person under $8$ will end up on horse that does not moves up or down.

Answer to Problem 20PPS

The probability is $\frac{2}{3}$ .

Explanation of Solution

Given:

  $6$ of $9$ people randomly filling the carousel.

  

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 total number of arrangements of the $9$ people is $\frac{9!}{6!}$ or $504$ . The number of favorable outcomes is the number of distinguishable permutations of the other eight places if a child under $8$ is on the horse that does not move up or down $\frac{8!}{5!}$ or $336$ . Calculate the probability.

  $P\left(\text{child under }8\text{ on broken horse}\right)=\frac{336}{504}\text{ or }\frac{2}{3}.$

Hence, the probability is $\frac{2}{3}$ .