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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?

xan
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2 Answers2

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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?

Ross Millikan
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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 –  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 –  Oct 14 '10 at 23:39
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    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
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In the word "mathematics", the the vowels are {a, e, a, i} and the consonants are {m, t, h, n, t, c, s}.

We will put gaps between and around the consonants, like so: _ m _ t _ h _ n _ t _ c _ s _

The number of ways the consonants can be permuted are: 7!/2! = 2,520 (we divide by 2! as the letter "t" is repeated)

Now, the four vowels need to be placed in 8 gaps. The first vowel can be placed in any of the 8 gaps. The second one, let's say, is forced to be with the first one to pair them up. The third one has 7 gaps. The fourth one is also forced to be with the third one. Therefore, there are 8 * 1 * 7 * 1 = 56 ways to permute the vowels. However, since "a" is repeated, we will divide by 2, arriving at 28 ways.

Combining the two results, there are: 28 * 2,520 = 70,560 ways to permute this.

Tanish Bengani
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