Chapter 12.4, Problem 31PPS

a.

To determine

To find: the combination that the person can choose 4 letters out of 12 different letters.

Answer to Problem 31PPS

The person can choose 4 letters out of 12 different letters in 495 ways.

Explanation of Solution

Given information:

Total number of letters is 12.

The person chose 4 letters out of 12.

Formula used :

  $c(n,r)=\frac{n!}{(n-r)!r!}$

Calculation:

Number of ways that the person can choose 4 letters out of 12 different letters are-

  $\begin{array}{l}c(n,r)=\frac{n!}{(n-r)!r!}\\ c(12,4)=\frac{12!}{(12-4)!4!}\left[\text{Since }n=12\text{ and }r=4\right]\\ =\frac{12!}{8!4!}\\ =\frac{12\times 11\times 10\times 9\times 8!}{4\times 3\times 2\times 1\times 8!}\left[\text{Divide by common factor}\text{.}\right]\\ =495\text{ ways}\end{array}$

Hence, there are 495 ways that the person can choose 4 letters out of 12 different letters.

b.

To determine

To find: the permutation that the person will arrange the three letters out of 4.

Answer to Problem 31PPS

The person will arrange the three letters out of 4 in 24 arrangements.

Explanation of Solution

Given information:

Total number of letters is 4.

The person chose 3 letters out of 4.

Formula used :

  $P(n,r)=\frac{n!}{(n-r)!}$

Calculation:

Number of arrangements that the person will arrange the three letters out of 4 are-

  $\begin{array}{l}P(n,r)=\frac{n!}{(n-r)!}\\ P(4,3)=\frac{4!}{(4-3)!}\left[\text{Since }n=4\text{ and }r=3\right]\\ =\frac{4!}{1!}\\ =24\text{ arrangements}\end{array}$

Hence, there are 24 arrangements that the person will arrange the three letters out of 4.

c.

To determine

To find: the words that the arrangement of the 3 letters will make.

Answer to Problem 31PPS

There are 9 words to make the arrangement of 3 letters.

Explanation of Solution

Given information:

Total number of letters is 4- A, T, R and E

We have to arrange three letters to make a word.

The words are-

ART, ATE, ARE, TAR, TEA, RAT, EAT, EAR, and ERA.

Hence, there are 9 words to make the arrangement of 3 letters.