SOLUTION: The letters of DIRECTOR are arranged at random. What is the probability of vowels occupying ONLY even places. Kindly provide the detailed explanation.

Question 472787: The letters of DIRECTOR are arranged at random. What is the probability of vowels occupying ONLY even places.
Kindly provide the detailed explanation.
Answer by Edwin McCravy(20054) (Show Source): You can put this solution on YOUR website!
The letters of DIRECTOR are arranged at random. What is the probability of vowels occupying ONLY even places.
Kindly provide the detailed explanation.

DIRECTOR has 5 consonants (C) and 3 vowels (V).  There are 4 even places 
but only 3 vowels.  So we choose 3 of the 4 even places to put the 3
vowels.  So we can chose 3 of the 4 even places to  

So it has one of these C(4,3) forms:

CVCVCVCC, CVCVCCCV, CVCCCVCV, or CCCVCVCV

Every one of these C(4,3) ways to place the 3 vowels leaves the 
remaining 5 consonants D,R,C,T,R
to arrange around them.  The 2 R's are indistinguishable, so for each 
of the C(4,1) ways there are  ways to rearrange D,R,C,T,R
around the three vowels. So the numerator of the probabiliy is:

C(4,3)* = 240

The denominator is the number of ways to arrange the letters DIRECTOR
in any manner, with the 2 R's indistinguishable:

 = 20160

So the desired probability is  = 1/84


Edwin

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