# How many different arrangements can be formed of the word "equation" if all the vowels are to be kept together?

#### Asawira Emaan

##### New member
Asalamoalikum,

Kindly can someone solve these with explanation?

1. In how many ways can four French books, two English books and three German books be arranged on a shelf so that all books in same language are together.

2. How many different arrangements can be formed of the word "equation" if all the vowels are to be kept together?

3. A combination lock has five wheels, each labeled with the ten digits from 0 to 9. How many five number opening combinations are possible,
(i) assuming no digit is repeated.
(ii)assuming digits can be repeated.

Thank you.

#### Country Boy

##### Well-known member
MHB Math Helper
Asalamoalikum,

Kindly can someone solve these with explanation?
You won't learn anything by people doing your homework for you! Here are some hints:

1. In how many ways can four French books, two English books and three German books be arranged on a shelf so that all books in same language are together.
First, treat the books of each language, which must be kept together, as one object. In how many ways can you arrange 3 objects? Now, in how many ways can you arrange just the 4 French books? The two English books? The three German books? By the "fundamental counting principle", multiply those.

2. How many different arrangements can be formed of the word "equation" if all the vowels are to be kept together?
Treat the vowels, e, u a, i, and o, that are to be kept together, as a single letter. there are also 3 consonants so, treating the vowels as a single letter, there are 4 objects. In how many ways can you arrange 4 objects. In how many ways can you arrange the 5 vowels?

3. A combination lock has five wheels, each labeled with the ten digits from 0 to 9. How many five number opening combinations are possible,
(i) assuming no digit is repeated.
There are 10 choices for the first digit, then 9 for the second, etc.

(ii)assuming digits can be repeated.
Then there are 10 choices for every digit.

Thank you.

#### Asawira Emaan

##### New member
You won't learn anything by people doing your homework for you! Here are some hints:

First, treat the books of each language, which must be kept together, as one object. In how many ways can you arrange 3 objects? Now, in how many ways can you arrange just the 4 French books? The two English books? The three German books? By the "fundamental counting principle", multiply those.

Treat the vowels, e, u a, i, and o, that are to be kept together, as a single letter. there are also 3 consonants so, treating the vowels as a single letter, there are 4 objects. In how many ways can you arrange 4 objects. In how many ways can you arrange the 5 vowels?

There are 10 choices for the first digit, then 9 for the second, etc.

Then there are 10 choices for every digit.