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In how many ways can the letters of word "EDUCATION" be arranged such

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In how many ways can the letters of word "EDUCATION" be arranged such  [#permalink]

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New post Updated on: 08 Jul 2015, 08:03
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In how many ways can the letters of word "EDUCATION" be arranged such that NO two Consonants (C,D,T,N) appear together?

A) 9!
B) 5!*4!
C) 15*5!*4!
D) 5!*4!*2!
E) 6!*4!

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Originally posted by GMATinsight on 08 Jul 2015, 08:01.
Last edited by Bunuel on 08 Jul 2015, 08:03, edited 1 time in total.
Renamed the topic and edited the question.
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Re: In how many ways can the letters of word "EDUCATION" be arranged such  [#permalink]

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New post 08 Jul 2015, 08:32
GMATinsight wrote:
In how many ways can the letters of word "EDUCATION" be arranged such that NO two Consonants (C,D,T,N) appear together?

A) 9!
B) 5!*4!
C) 15*5!*4!
D) 5!*4!*2!
E) 6!*4!


Tricky question.

There are 2 cases possible to have NO 2 consonants together.

Case 1: VCVCVCVCV. We can this arrangement in 5!*4! ways (look at in-how-many-ways-can-the-letters-of-word-education-be-arranged-such-201428.html#p1546972).

Case 2: VVCVCVCVC. Number of arrangements possible = 8! (all vowels and consonants are different!) (though still not sure why we shouldnt be mutiplying this by 2 for the {VV} group!)

Thus total number of ways = 8!+5!*4! = 15*5!*4!
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Re: In how many ways can the letters of word "EDUCATION" be arranged such  [#permalink]

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New post 08 Jul 2015, 09:14
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GMATinsight wrote:
In how many ways can the letters of word "EDUCATION" be arranged such that NO two Consonants (C,D,T,N) appear together?

A) 9!
B) 5!*4!
C) 15*5!*4!
D) 5!*4!*2!
E) 6!*4!


Another Method

__V__V__V__V__V__
Here is the arrangement of 5 vowels such that between any two vowels at most one Consonant can come but since the consonants can also appear before the leftmost vowel or after the rightmost vewel so it gives us 6 places available for consonants (represented by “___”)

But since we require only 4 places due to 4 consonants to be arranged so we will select the 4 places out of 6 in 6C4 ways = 15 ways

In every selected the Consonants can be arranged in 4! Ways at the selected 4 places

All the vowels can also be arranged among themselves in 5! Ways

so Total ways to arrange the letters as per desired condition = 15*5!*4!

Answer: Option C
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Re: In how many ways can the letters of word "EDUCATION" be arranged such  [#permalink]

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New post 23 Jul 2018, 11:08
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Re: In how many ways can the letters of word "EDUCATION" be arranged such   [#permalink] 23 Jul 2018, 11:08
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