In the below solved problem, every thing is okay, but if we have $4$ consonants then why we are giving $5!$? and is this a combination problem? how to distinguish?
Question:
In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?
Answer:
The word 'OPTICAL' contains $7$ different letters.
When the vowels OIA are always together, they can be supposed to form one letter.
Then, we have to arrange the letters PTCL (OIA).
Now, $5$ letters can be arranged in $5! = 120$ ways.
The vowels (OIA) can be arranged among themselves in $3! = 6$ ways.
Required number of ways $= (120*6) = 720$.
Best Answer
There are $4$ consonants and $1$ group of vowels, so there are $5$ elements to permute. Yes, this is a combinatorial problem because we are counting the number of possibilities that satisfy certain conditions.