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In how many different ways

This topic has 2 expert replies and 0 member replies

In how many different ways

Post 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

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GMAT/MBA Expert

Top Reply
Post Thu Dec 07, 2017 7:49 am
Another way to think about this problem is as two separate permutations.

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

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Post 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)

Answer: D
--------------------------

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/what-should-be-the-answer-t267256.html
- 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/no-two-ladies-sit-together-t275661.html
- http://www.beatthegmat.com/combinations-t123249.html


Cheers,
Brent

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Brent Hanneson – Founder of GMATPrepNow.com
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