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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 $= (120*6) = 720$.

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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.

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

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