Chapter 11.1, Problem 20PPSE

Solution

(a)

To calculate: The number of arrangements of four tires on a car.

The number of arrangements of four tires on a car is 24.

Given:

Total number of tires on a car $=4$

Concept used:

The number of arrangements of r objects from a total of n objects, when order of the arrangement matters is calculated using permutations $n_{P_r}=\frac{n!}{\left(n-r\right)!}$

Calculation:

The total number of tires on a car $n=4$

The total number of tires to be arranged $r=4$

The total number of ways four car tires can be arranged on a car is calculated as shown below

  $\begin{array}{c}{n}_{P_r}=\frac{n!}{\left(n-r\right)!}\\ =\frac{4!}{\left(4-4\right)!}\\ =\frac{4!}{0!}\\ =24\end{array}$

The number of ways four tires can be arranged on a car is 24.

(b)

To calculate: The number of arrangements of four tires on a car when a spare tire is included.

The number of arrangements of four tires on a car when a spare tire is included is 120.

Given:

Total number of tires on a car including spare tire $=5$

Calculation:

The total number of tires on a car including spare tire $n=5$ .

The total number of tires to be arranged $r=4$ .

The total number of ways four car tires can be arranged on a car is calculated as shown below:

  $\begin{array}{c}{n}_{P_r}=\frac{n!}{\left(n-r\right)!}\\ =\frac{5!}{\left(5-4\right)!}\\ =\frac{5!}{1!}\\ =120\end{array}$

The number of ways four tires can be arranged on a car when spare tire is included is 120.