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I had tried the question and got the answer 11!/(6!2!) but the answer given is 11!/6! if any body can explain that why 2! is not in the answer or the answer is wrong.

Nikita
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    Perhaps the consecutive double T's has to do something. – CAGT Apr 01 '16 at 13:33
  • Welcome to Math.SE. Please review [ask], and in particular try to make the body of your Question as self-contained as possible (not relying on the title alone to state a problem). Note that if the consonants must be in alphabetic order, the only difficulty is with placement of vowels. The vowels are all distinct in this setting. – hardmath Apr 01 '16 at 13:36

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Well, what are the consonants? They are P, R, M, T, T, and N. They must be arranged in the following order: M, N, P, R, T, and T (notice the Ts are indistinct). We can now use distributions to determine the vowel-consonant order. Treating the consonants as indistinct (there's only one way to place them), we have $\dbinom{11}{6}$ ways to arrange. There are now $5!$ ways to arrange the vowels, as there are no constraints on vowels. The answer is $\boxed{\dbinom{11}{6} \times 5!}.$

K. Jiang
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So you have MNPRTT, and you need to "stick" each one of AEIOU at any location.

In other words, all you need to do is:

  • Choose $5$ out of the $11$ available slots
  • Permute the $5$ letters AEIOU in any order

The total number of ways to do it is:

$$\binom{11}{5}\cdot5!=\frac{11!}{6!}$$

barak manos
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  • @Nikita: You can only accept one answer, and I've noticed that the other answer (which you have originally accepted) was there first, and is also quite similar to my answer. So unless you think that my answer is favorable over the other one, you should stick to your previous choice :) – barak manos Apr 01 '16 at 13:52
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    Well, second thought, the statement "we have $\binom{11}{6}$ to arrange" is wrong in the context of this question, even though it "somehow" gets the correct answer, as $\binom{11}{6}=\binom{11}{5}$. – barak manos Apr 01 '16 at 13:56
  • "$\binom{11}6$ ways to arrange" has been taken by Jiang for first fixing the consonants. – true blue anil Apr 01 '16 at 16:07
  • @trueblueanil: Yep, I get it... So it's more or less the same answer, and I'll leave it for OP to decide... Thanks. – barak manos Apr 01 '16 at 16:18