Chapter 10.5, Problem 7E

(a)

To determine

To calculate:

All of the letters in the given word

  $FAMILY$

Answer to Problem 7E

All of the letters in the given word $FAMILY$ is $720$ .

Explanation of Solution

Given information:

  $FAMILY$

Calculation:

The formula for permutation for $n$ objects is:

  $P{}_n=n!$

Where,

  $n=6$

There are total six words in this letter $FAMILY$ .

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

  $\begin{array}{l}P{}_6=6!\\ P{}_6=6\times 5\times 4\times 3\times 2\times 1\\ P{}_6=720\end{array}$

Hence, all of the letters in the given word $FAMILY$ is $720$ .

(b)

To determine

To calculate:

Two of the letter in this word

  $FAMILY$

Answer to Problem 7E

The two of the letters in the given word $FAMILY$ is $30$ .

Explanation of Solution

Given information:

  $FAMILY$

Calculation:

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

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

Where,

  $\begin{array}{l}n=6\\ r=2\end{array}$

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

  $\begin{array}{l}P{}_2=\frac{6!}{\left(6-2\right)!}\\ P{}_2=\frac{6!}{4!}\\ P{}_2=\frac{6\times 5\times 4\times 3\times 2\times 1}{4\times 3\times 2\times 1}\\ P{}_2=30\end{array}$

Hence, two of the letters in the given word $FAMILY$ is $30$ .