Chapter 10.5, Problem 6E

(a)

To determine

To calculate:

All of the letters in the given word

  $WATER$

Answer to Problem 6E

All of the letters in the given word $WATER$ is $120$ .

Explanation of Solution

Given information:

  $WATER$

Calculation:

The formula for permutation for $n$ objects is:

  $P{}_n=n!$

Where,

  $n=5$

There are total five words in this letter $WATER$ .

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

  $\begin{array}{l}P{}_5=5!\\ P{}_5=5\times 4\times 3\times 2\times 1\\ P{}_5=120\end{array}$

Hence, all of the letters in the given word $WATER$ is $120$ .

(b)

To determine

To calculate:

Two of the letter in this word

  $WATER$

Answer to Problem 6E

The two of the letters in the given word $WATER$ is $20$ .

Explanation of Solution

Given information:

  $WATER$

Calculation:

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

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

Where,

  $\begin{array}{l}n=5\\ r=2\end{array}$

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

  $\begin{array}{l}P{}_2=\frac{5!}{\left(5-2\right)!}\\ P{}_2=\frac{5!}{3!}\\ P{}_2=\frac{5\times 4\times 3\times 2\times 1}{3\times 2\times 1}\\ P{}_2=20\end{array}$

Hence, two of the letters in the given word $WATER$ is $20$ .