# 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?

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.

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