• Award-winning private GMAT tutoring
Register now and save up to $200 Available with Beat the GMAT members only code • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

Available with Beat the GMAT members only code

• 5-Day Free Trial
5-day free, full-access trial TTP Quant

Available with Beat the GMAT members only code

• Magoosh
Study with Magoosh GMAT prep

Available with Beat the GMAT members only code

## In how many different ways

This topic has 2 expert replies and 0 member replies
BTGmoderatorDC Moderator
Joined
07 Sep 2017
Posted:
710 messages
Followed by:
4 members

#### In how many different ways

Thu Dec 07, 2017 5:37 am
In how many different ways can the letters of the word 'DETAIL' be arranged such that the vowels must occupy only the odd positions?

A. None of these

B. 64

C. 120

D. 36

E. 360

I'm quite confused, can some experts help me find the best solution in this?

OA D

### GMAT/MBA Expert

ErikaPrepScholar Legendary Member
Joined
20 Jul 2017
Posted:
503 messages
Followed by:
7 members
86
GMAT Score:
770
Thu Dec 07, 2017 7:49 am

We know that the odd spaces (1, 3, and 5) must contain our vowels. We have three different vowels (E, A, and I). Thus there are 3! (or 6) possible orders for the vowels in spaces 1, 3, and 5.

We know that the even spaces (2, 4, and 6) must contain our consonants. We have three different consonants (D, T, and L). Thus there are 3! (of 6) possible orders for the consonants in spaces 2, 4, and 6.

To figure out all possible combinations of vowel orders in the odd spaces and consonant orders in the even spaces, we multiply the two together:

3! * 3! = 6 * 6 = 36

_________________

Erika John - Content Manager/Lead Instructor
https://gmat.prepscholar.com/gmat/s/

Get tutoring from me or another PrepScholar GMAT expert: https://gmat.prepscholar.com/gmat/s/tutoring/

Learn about our exclusive savings for BTG members (up to 25% off) and our 5 day free trial

Check out our PrepScholar GMAT YouTube channel, and read our expert guides on the PrepScholar GMAT blog

### GMAT/MBA Expert

Brent@GMATPrepNow GMAT Instructor
Joined
08 Dec 2008
Posted:
11499 messages
Followed by:
1229 members
5254
GMAT Score:
770
Thu Dec 07, 2017 7:32 am
lheiannie07 wrote:
In how many different ways can the letters of the word 'DETAIL' be arranged such that the vowels must occupy only the odd positions?

A. None of these

B. 64

C. 120

D. 36

E. 360
Vowels: E, A, I
Consonants: D, T, L

Spaces: #1, #2, #3, #4, #5, #6,

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

Stage 1: Select a vowel to go in space #1
There are 3 vowels to choose from, so we can complete stage 1 in 3 ways

Stage 2: Select a vowel to go in space #3
There are 2 remaining vowels from which to choose, so we can complete this stage in 2 ways.

Stage 3: Select a vowel to go in space #5
There is 1 remaining vowel from which to choose, so we can complete this stage in 1 ways.

Stage 4: Select a consonant to go in space #2
There are 3 consonants to choose from, so we can complete stage 4 in 3 ways .

Stage 5: Select a consonant to go in space #4
There are 2 remaining consonants from which to choose, so we can complete this stage in 2 ways.

Stage 6: Select a consonant to go in space #6
There is 1 remaining consonant from which to choose, so we can complete this stage in 1 ways.

By the Fundamental Counting Principle (FCP), we can complete all 6 stages (and thus arrange all of the letters) in (3)(2)(1)(3)(2)(1) ways (= 36 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

_________________
Brent Hanneson â€“ Founder of GMATPrepNow.com
Use our video course along with

Check out the online reviews of our course
Come see all of our free resources

GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months!

### Top First Responders*

1 GMATGuruNY 65 first replies
2 Rich.C@EMPOWERgma... 46 first replies
3 Brent@GMATPrepNow 41 first replies
4 Jay@ManhattanReview 24 first replies
5 Terry@ThePrinceto... 10 first replies
* Only counts replies to topics started in last 30 days
See More Top Beat The GMAT Members

### Most Active Experts

1 GMATGuruNY

The Princeton Review Teacher

129 posts
2 Rich.C@EMPOWERgma...

EMPOWERgmat

114 posts
3 Jeff@TargetTestPrep

Target Test Prep

98 posts
4 Scott@TargetTestPrep

Target Test Prep

92 posts
5 Max@Math Revolution

Math Revolution

89 posts
See More Top Beat The GMAT Experts