Chapter 10.5, Problem 5E

(a)

To determine

To calculate:

All of the letters in the given word

  $ROCK$

Answer to Problem 5E

All of the letters in the given word $ROCK$ is $24$ .

Explanation of Solution

Given information:

  $ROCK$

Calculation:

The formula for permutation for $n$ objects is:

  $P{}_n=n!$

Where,

  $n=4$

There are total three words in this letter $ROCK$ .

The number of the permutations to arrange all of letters is as:

  $\begin{array}{l}P{}_4=4!\\ P{}_4=4\times 3\times 2\times 1\\ P{}_4=24\end{array}$

Hence, all of the letters in the given word $ROCK$ is $24$ .

(b)

To determine

To calculate:

Two of the letter in this word

  $ROCK$

Answer to Problem 5E

The two of the letters in the given word $ROCK$ is $12$ .

Explanation of Solution

Given information:

  $ROCK$

Calculation:

The formula for permutation for $n$ objects taken $r$ at the time is as:

  $P{}_r=\frac{n!}{\left(n-r\right)!}$

Where,

  $\begin{array}{l}n=4\\ r=2\end{array}$

The number of the permutations to arrange two of letters is as:

  $\begin{array}{l}P{}_2=\frac{4!}{\left(4-2\right)!}\\ P{}_2=\frac{4!}{2!}\\ P{}_2=\frac{4\times 3\times 2\times 1}{2\times 1}\\ P{}_2=12\end{array}$

Hence, two of the letters in the given word $ROCK$ is $12$ .