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I am trying to solve the following problem, but I am a bit stuck. The question is as follows.

The language of a certain island has only the letters A, B, C, D, E. Every place name must start and end with a consonant, consist of exactly 6 letters, contain exactly 2 vowels, which may not be next to each other, and cannot have 2 consecutive copies of the same consonant. How many possible place names are there?

From what I have established, since the start and end of the place name has to end with consonants, there are $3^{2}$ = 9 possibilities for this.

This results in $\ ^{4}C_2$ different choices for placing the vowels.

However, I am a bit confused after this.

Any help would be greatly appreciated.

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1 Answer 1

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HINT: These names can only be of the forms CVCVCC, CCVCVC, and CVCCVC, where V represents a vowel and C a consonant, and consecutive consonants must be distinct. The CVCVCC and CCVCVC types are mirror images of each other, so there must be the same number of each. Thus, all you have to do is count the names of CVCVCC type, double it, and add the number of names of CVCCVC type.

  • CVCVCC: There are $3$ possibilities for the first letter, $2$ for the second, $3$ for the third, ...

  • CVCCVC: The same straightforward approach works here.

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All three patterns have the same number of possibilities –  Henry Nov 10 '13 at 12:30
    
@Henry: I know. I was rather hoping that the OP might spot that from the patterns. –  Brian M. Scott Nov 10 '13 at 12:31
    
@BrianM.Scott I totally understand it with your help. Thanks. –  user98380 Nov 10 '13 at 12:39
    
@user98380: You’re welcome –  Brian M. Scott Nov 10 '13 at 12:40

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