Chapter 10.5, Problem 43E

(a)

To determine

To explain: whether there are more permutations of all 4 or of 3 of the objects.

Answer to Problem 43E

The results of both permutations are same.

Explanation of Solution

Calculation:

The number of permutations of n objects taken r at a time, where $r\le n$ is given by

  $P{}_r=\frac{n!}{(n-r)!}$

Therefore, the number of permutations of 4 objects takes 4n at a time, where as follows

  $\begin{array}{l}P{}_4=\frac{4!}{(4-4)!}\\ P{}_4=24\end{array}$

Equivalently, the number of permutations 4 objects taken 3 at a time is as follows

  $\begin{array}{l}P{}_3=\frac{4!}{(4-3)!}\\ P{}_3=24\end{array}$

Since, $0!$ and $1!$ are both 1, the results are same.

Conclusion:

Therefore, the results of both permutations are same.

(b)

To determine

To explain: whether there are more combinations of all 4 or of 3 of the objects.

Answer to Problem 43E

There are more combinations for 4 objects taken 3 at a time.

Explanation of Solution

Calculation:

The number of combinations of n objects taken r at a time, where $r\le n$ is given by

  $C{}_r=\frac{n!}{r!(n-r)!}$

Therefore, the number of combinations of 4 objects takes 4n at a time, whereas follows

  $\begin{array}{l}C{}_4=\frac{4!}{4!(4-4)!}\\ C{}_4=1\end{array}$

Equivalently, the number of combinations of 4 objects taken 3 at a time is as follows

  $\begin{array}{l}C{}_3=\frac{4!}{3!(4-3)!}\\ C{}_3=4\end{array}$

The results are not same. There are more combinations for 4 objects taken 3 at a time.

Conclusion:

Therefore, there are more combinations for 4 objects taken 3 at a time.

(c)

To determine

To compare: The answers of part a and part b

Answer to Problem 43E

The results were similar in first case and different in later case.

Explanation of Solution

Calculation:

For, $n\succ 1$ , conclude the following from the preceding results. The conclusions are as follows

  $P{}_n=P{}_{n-1}$

On the other hands,

  $C{}_n\prec C{}_{n-1}$

This explains why the results were similar in first case and different in later case.

Conclusion:

Therefore, the results were similar in first case and different in later case.