[Math] How many arrangements of MATHEMATICS are there in which each consonant is adjacent to a vowel

combinatorics

How many arrangements of MATHEMATICS are there in which each consonant is adjacent to a vowel?

Now I have calculated it to be
$$
11 \cdot \frac{7!}{2!2!} \cdot \frac{4!}{2!}
$$
for 11 distinct distributions of consonants among vowels with repetitions making sure that no two vowels are together.
(If not, then at least one consonant will be not adjacent to any of the four vowels.)

I'm not sure about it and I would like to verify my answer. Any help will be appreciated.

Best Answer

With eight consonants there's only one way to distribute consonants and vowels (cvccvccvccvc) and removing one of them can only be done in five distinct ways (with vowels denoted by - for better visual appearance):

  1. -cc-cc-cc-c

  2. c-c-cc-cc-c

  3. c-cc-c-cc-c

  4. c-cc-cc-c-c

  5. c-cc-cc-cc-

    Then it's only to place the consonants in ${7!\over2!2!}$ ways (since the order of m's and t's doesn't matter) and vowels in ${4!\over2!}$ ways (since the order of a's doesn't matter).

The answer is: $5{7!\over2!2!}{4!\over2!}$

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