Question

Use the method illustrated in the solutions to Exercise 9.2.39 to answer the following questions. (a) How many ways can the letters of the word DANCER be arranged in a row? Since the letters in the given word are distinct, there are as many arrangements of these letters in a row as there are permutations of a set with elements. So the answer is . (b) How many ways can the letters of the word DANCER be arranged in a row if D and A must remain together (in order) as a unit? (c) How many ways can the letters of the word DANCER be arranged in a row if the letters NCE must remain together (in order) as a unit?

Answer 1

Answer:

(a) 720 ways

(b) 120 ways

(c) 24 ways

Step-by-step explanation:

Given

$Word = DANCER$

$n =6$ --- number of letters

Solving (a): Number of arrangements.

We have:

$n =6$

So, the number of arrangements is calculated as:

$Total =n!$

This gives:

$Total =6!$

This gives:

$Total =6*5*4*3*2*1$

$Total =720$

Solving (b): DA as a unit

DA as a unit implies that, we have:

[DA] N C E R

So, we have:

$n = 5$

So, the number of arrangements is calculated as:

$Total =n!$

This gives:

$Total =5!$

This gives:

$Total =5*4*3*2*1$

$Total =120$

Solving (c): NCE as a unit

NCE as a unit implies that, we have:

D A [NCE] R

So, we have:

$n = 4$

So, the number of arrangements is calculated as:

$Total =n!$

This gives:

$Total =4!$

This gives:

$Total =4*3*2*1$

$Total =24$