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

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CEO
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Joined: 08 Jul 2010
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WE: Education (Education)
In how many ways can the letters of word "EDUCATION" be arranged such  [#permalink]

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Updated on: 08 Jul 2015, 08:03
10
00:00

Difficulty:

85% (hard)

Question Stats:

46% (02:19) correct 54% (02:16) wrong based on 117 sessions

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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.
CEO
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WE: Engineering (Aerospace and Defense)
Re: In how many ways can the letters of word "EDUCATION" be arranged such  [#permalink]

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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!
CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2959
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
Re: In how many ways can the letters of word "EDUCATION" be arranged such  [#permalink]

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08 Jul 2015, 09:14
3
1
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!

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

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23 Jul 2018, 11:08
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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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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