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

(a) 720 ways

(b) 120 ways

(c) 24 ways

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`