# In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?

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 $= (120*6) = 720$.

## 1 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.

• Thanks fahrbach. One thing more.. Can I find the combination of word MATHEMATICS where r choices are not given? – user160285 Jun 26 '14 at 17:03
• @user160285, I don't understand what you're asking. My guess is that you can find it though. – fahrbach Jun 26 '14 at 17:07
• I wanted to know that can we find the number of combinations of the letters of the word "MATHEMATICS"? – user160285 Jun 26 '14 at 17:09
• Oh, all of the permutations (anagrams) of the word "MATHEMATICS"? So, "MATHEMATICS", "AMTHEMATICS", etc.? Yes, you can. – fahrbach Jun 26 '14 at 17:12
• No i didn't ask that, actually i am asking a foolish thing. I know. Well thanks fahrbach.. – user160285 Jun 26 '14 at 17:15