Chapter 13.2, Problem 37E

(a)

To determine

To find: The number of different starting line-up based on manufacturer are possible.

Answer to Problem 37E

The number of different line-up are $7.85\times {10}^{17}$ .

Explanation of Solution

Given:

There are $20F$ cars, $14C$ cars, and $9P$ cars.

Calculation:

This is the permutation with repetition of $43$ elements $\left(20+14+9=43\right)$ .

The possible numbers of arrangements are as follows.

  $\frac{43!}{20!14!9!}=7.85\times {10}^{17}$

Therefore, the number of different line-up are $7.85\times {10}^{17}$ .

(b)

To determine

To rearrange: The number of ways to arrange the cars on the track.

Answer to Problem 37E

The numbers of possible arrangements are $42!$ .

Explanation of Solution

Given:

There are $20F$ cars, $14C$ cars, and $9P$ cars.

Calculation:

The track has oval shape, so the problem describes a circular permutation.

The numbers of possible arrangements are as follows.

  $\begin{array}{c}\left(n-1\right)!=\left(43-1\right)!\\ =42!\end{array}$

Therefore, the numbers of possible arrangements are $42!$ .

(c)

To determine

To rearrange: The number of ways to arrange the cars on the track relative to the leader of the track.

Answer to Problem 37E

The numbers of possible arrangements are $43!$ .

Explanation of Solution

Given:

There are $20F$ cars, $14C$ cars, and $9P$ cars.

Calculation:

In this case, the permutation becomes linear.

The numbers of possible arrangements are as follows.

  $n!=43!$

Therefore, the numbers of possible arrangements are $43!$ .