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# In how many ways can the letters of the word "permutations"

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Director
Joined: 03 Jul 2003
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In how many ways can the letters of the word "permutations" [#permalink]  07 Jan 2004, 08:20
In how many ways can the letters of the word "permutations" be arranged such that order of vowels remains unchanged?
Senior Manager
Joined: 05 May 2003
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What is the official answer ??
SVP
Joined: 30 Oct 2003
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P E R M U T A T I O N S
x E x x U x A x I O x x

If we dont want to change the order then there 4 possible ways for EUAIO to occupy 12 places as follows

ExxUxAxIOxxx
xExxUxAxIOxx
xxExxUxAxIOx
xxxExxUxAxIO

Among available 7 places we need to arrange 7 letters with letter T repeating twice.
So we have 7!/2!
Total ways = 4 * 7! / 2! = 10080

What is the official answer ?
Senior Manager
Joined: 22 May 2003
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The problem just says that the order of the vowels must be respected. So it could be euaioxxxxxxx, xeuaioxxxxxx, xexuxaxixoxx, etc...

In that case I think the answer 12! divided by 2! (because of the double t) and 5! (to eliminate the different orders of the 5 vowels).

That is 12!/(2!*5!)
SVP
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I agree with you martin.
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