Question

Form a seven-letter word by mixing up the letters in the word FIXTURE.

How many ways can you do this if no vowel is isolated between two consonants?

Answer 1

You have the main rule: no vowel is isolated between two consonants.

The word FIXTURE consists of 4 consonants and 3 vowels.

There are such possible cases:

1. Formed word begins with three vowels and ends with 4 consonants.

The number of such words is
`3!\cdot 4!=6\cdot 24=144.`

2. Formed word begins with two vowels and ends with one vowel (between them stand all consonants).

The number of such words is
`3\cdot 2!\cdot 4!=6\cdot 24=144.`

3. Formed word begins with one vowel and ends with two vowels (between them stand all consonants).

The number of such words is
`3\cdot 2!\cdot 4!=6\cdot 24=144.`

4. Formed word begins with with 4 consonants and ends with 3 vowels.

The number of such words is
`3!\cdot 4!=6\cdot 24=144.`

5. In total 144+144+144+144=576 different words.

Answer 2

There are 4 consonants and 3 vowels. If the vowels can't be isolated between two consonants then the vowels would have to be right next to each other. Let's consider the vowels as a group so that they can be arranged in 3!=6 ways and the consonants 4!=24 ways. So:
6x24=144ways