Permutations and Combination rearrangement

I approached this exam question the wrong way apparently, help please?

Consider the word "mathematics". In how many ways can you rearrange all the letters so that the vowels are paired and always apart?

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How did you try it? And what does "paired and always apart" mean? – Paul VanKoughnett Oct 14 '10 at 23:30
Does "rearrange all the letters" imply there are no fixed points? – Douglas S. Stones Oct 14 '10 at 23:37

1 Answer

Given there are four vowels, list all the ways to split them into pairs. Now for each set of pairs think of each pair as a single character. How many ways can you permute the resulting collection? How many of those have the vowel pairs next to each other?

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I did, I also divided out the repeating "a" as well as doing the same with the remaining letters of M,M,T,H,C,S. 1836 was not the answer – Kalena Oct 14 '10 at 23:38
paired as in ae are a pair or ai BUT they cannot be near another pair. and Yes, no fixed points – Kalena Oct 14 '10 at 23:39
The no fixed points may be the problem. Frequently if you are asked how many ways can you rearrange abc the expected answer is 6, not 2. – Ross Millikan Oct 14 '10 at 23:51