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The question is exactly this:

How many orders are there of the letters in the word M E D I T E R R A N E A N? In how many of these are the vowels consecutive?

(Vowels can be consecutive in 2,3,4,5,6)

I managed to find the answer for the first question, but stuck on the second one.

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Hint for the second one: since all vowels have to be consecutive, you can think of them as forming a single block. So you need to find the number ways to rearrange the letters "M D T R R N N V", where "V" stands for the block of the vowels as a single entity. But that won't be the final answer. Inside that block of vowels, those vowels can be arranged in all different possible ways. So you need to multiply by the number of ways to arrange "E I E A E A".

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  • $\begingroup$ I get what you mean, but do you not have to take account for the positions of vowels? For example, the times that vowels are consecutive between consonants so "M D E I T R R N N . . .". Because I'm assuming the question is asking for any arrangements with any kind e.g,(2,3,4,5,6) of consecutive vowels. $\endgroup$ – christopher chong Sep 11 '17 at 5:39
  • $\begingroup$ The wording of the question suggests that all vowels are consecutive, as a single group of all vowels -- at least that's how I understood the question. A different interpretation is also possible, I guess, but in that case I need to think again about it. And maybe you could clarify with the source of this question (your teacher?) what they actually mean. $\endgroup$ – zipirovich Sep 11 '17 at 5:56
  • $\begingroup$ I can tell why it can be confusing, but I'm very certain the question suggests that the consecutive vowels can be in 2-6's. Thanks though! $\endgroup$ – christopher chong Sep 11 '17 at 6:14
  • $\begingroup$ @christopherchong The question is asking in how many arrangements are all the vowels consecutive. Otherwise, it would have asked in how many arrangements at least two of the vowels are consecutive. $\endgroup$ – N. F. Taussig Sep 11 '17 at 8:03
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To find the options in which no vowels are consecutive, you can first find the consonant-vowel layout like VCVCVCVCCVCCVCV, which has $\binom 86$ options from the $6$ gaps plus $2$ end positions $= 8 $ places for the $6$ vowels to be non-consecutive in a string of $7$ consonants. After that is selected, multiply by the vowel orderings and consonant orderings, which are the multinomial coefficients $\binom 6{3,2,1}$ and $\binom 7{2,2,1,1,1}$ respectively, giving

$$\binom 86\binom 6{3,2,1}\binom 7{2,2,1,1,1}=\frac{8!}{6!\,2!} \frac{6!}{3!\,2!}\frac{7!}{2!\,2!} = \frac{(7!)^2}{12}$$

options for no consecutive vowels. Then to find options which include some consecutive vowels, you can just subtract this from the total you have already found.

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