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## In how many ways can the letters of word "EDUCATION&quo

tagged by: AAPL

This topic has 2 expert replies and 0 member replies

### Top Member

AAPL Moderator
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#### In how many ways can the letters of word "EDUCATION&quo

Thu Nov 09, 2017 12:15 pm
In how many ways can the letters of word "EDUCATION" be arraged such all the vowels always appear together?

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

The OA is C.

Please, can any expert assist me with this PS question? I don't have it clear and I appreciate if any explain it for me. Thanks.

Last edited by AAPL on Sat Nov 11, 2017 11:02 am; edited 1 time in total

### GMAT/MBA Expert

DavidG@VeritasPrep Legendary Member
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Thu Nov 09, 2017 1:26 pm
AAPL wrote:
In how many ways can the letters of word "EDUCATION" be arraged such all the vowels always appear together?

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

The OA is C.

Please, can any expert assist me with this PS question? I don't have it clear and I appreciate if any explain it for me. Thanks.
First, link the vowels together and treat them as a single letter, so we have E-U-A-I-O as one letter and then four more: D, C, T, N, giving us a total of 5 "letters." The number of ways that we can arrange 5 letters is simply 5!.
But now we can also change the order of our vowels. There are five vowels, and therefore 5! ways that we could arrange those linked values.

So there's 5! ways to arrange the five "letters" and 5! ways that we can arrange the linked vowels, giving us a total of 5!* 5! possibilities. The answer is C

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### GMAT/MBA Expert

Brent@GMATPrepNow GMAT Instructor
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Thu Dec 07, 2017 7:13 am
AAPL wrote:
In how many ways can the letters of word "EDUCATION" be arranged such all the vowels always appear together?

A) 9!
B) 5!*4!
C) 5!*5!
D) 5!*4!*2!
E) 6!*4!
Vowels: E, U, A, I, O
Consonants: D, C, T, N

Take the task of arranging the letters and break it into stages.

Stage 1: Arrange the 5 vowels
We can arrange n unique objects in n! ways
So, we can arrange the 5 vowels in 5! ways
In other words, we can complete stage 1 in 5! ways

IMPORTANT: Once we've arranged the 5 vowels, we'll "glue" them together to create ONE "super-letter"
This will ensure that the 5 vowels appear together.

Stage 2: Arrange D, C, T, N, and the "super letter"
There are 5 unique objects to arrange.
So, we can arrange them in 5! ways

By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus arrange all 9 letters digits) in (5!)(5!) ways.

--------------------------

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: http://www.gmatprepnow.com/module/gmat-counting/video/775

You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat-counting/video/776

Then you can try solving the following questions:

EASY
- http://www.beatthegmat.com/counting-problem-company-recruitment-t244302.html
- http://www.beatthegmat.com/picking-a-5-digit-code-with-an-odd-middle-digit-t273110.html
- http://www.beatthegmat.com/permutation-combination-simple-one-t257412.html
- http://www.beatthegmat.com/simple-one-t270061.html

MEDIUM
- http://www.beatthegmat.com/combinatorics-solution-explanation-t273194.html
- http://www.beatthegmat.com/arabian-horses-good-one-t150703.html
- http://www.beatthegmat.com/sub-sets-probability-t273337.html
- http://www.beatthegmat.com/combinatorics-problem-t273180.html
- http://www.beatthegmat.com/digits-numbers-t270127.html
- http://www.beatthegmat.com/doubt-on-separator-method-t271047.html
- http://www.beatthegmat.com/combinatorics-problem-t267079.html

DIFFICULT
- http://www.beatthegmat.com/wonderful-p-c-ques-t271001.html
- http://www.beatthegmat.com/permutation-and-combination-t273915.html
- http://www.beatthegmat.com/permutation-t122873.html
- http://www.beatthegmat.com/combinations-t123249.html

Cheers,
Brent

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