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# In how many different ways can the letters of the word

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Manager
Joined: 30 Mar 2013
Posts: 109
In how many different ways can the letters of the word  [#permalink]

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23 Sep 2014, 03:41
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Difficulty:

15% (low)

Question Stats:

76% (01:08) correct 24% (01:39) wrong based on 169 sessions

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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
Manager
Joined: 30 Mar 2013
Posts: 109
Re: In how many different ways can the letters of the word  [#permalink]

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23 Sep 2014, 03:45
what i don't understand is why we don't have 4 positions for the vowels? 1 D 3 T 5 L 7. So here we have 3P3 for consonants, and 4P3 for vowels. This is, after all, how we did this question:
in-how-many-different-ways-can-the-letters-of-the-word-corporation-185090.html

thank you!
Math Expert
Joined: 02 Sep 2009
Posts: 50622
Re: In how many different ways can the letters of the word  [#permalink]

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23 Sep 2014, 03:52
usre123 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

OEOEOE

3 vowels EOI can be arranged on odd places in 3!=6 ways. Similarly 3 consonants can be arranged on even places in 3!=6 ways.

Total = 6*6 = 36.

P.S. This question cannot be from any reliable GMAT source. Please do not post questions from that source again. Thank you.
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Joined: 30 Mar 2013
Posts: 109
Re: In how many different ways can the letters of the word  [#permalink]

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23 Sep 2014, 03:58
Bunuel wrote:
usre123 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

OEOEOE

3 vowels EOI can be arranged on odd places in 3!=6 ways. Similarly 3 consonants can be arranged on even places in 3!=6 ways.

Total = 6*6 = 36.

P.S. This question cannot be from any reliable GMAT source. Please do not post questions from that source again. Thank you.

Ok, I won't, thank you.
But could you pls explain why we don't consider the one odd position after the 6th even one? just like we did in the link I posted in my answer above?
You can delete this questions if you wish. I don't wan to confuse the other people preparing for the exam.
Math Expert
Joined: 02 Sep 2009
Posts: 50622
Re: In how many different ways can the letters of the word  [#permalink]

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23 Sep 2014, 05:26
usre123 wrote:
Bunuel wrote:
usre123 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

OEOEOE

3 vowels EOI can be arranged on odd places in 3!=6 ways. Similarly 3 consonants can be arranged on even places in 3!=6 ways.

Total = 6*6 = 36.

P.S. This question cannot be from any reliable GMAT source. Please do not post questions from that source again. Thank you.

Ok, I won't, thank you.
But could you pls explain why we don't consider the one odd position after the 6th even one? just like we did in the link I posted in my answer above?
You can delete this questions if you wish. I don't wan to confuse the other people preparing for the exam.

We have 6 letters, so only 6 places. There is no 7th place.
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Manager
Joined: 30 Mar 2013
Posts: 109
Re: In how many different ways can the letters of the word  [#permalink]

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23 Sep 2014, 09:35
OEOEOE

3 vowels EOI can be arranged on odd places in 3!=6 ways. Similarly 3 consonants can be arranged on even places in 3!=6 ways.

Total = 6*6 = 36.

P.S. This question cannot be from any reliable GMAT source. Please do not post questions from that source again. Thank you.[/quote]

Ok, I won't, thank you.
But could you pls explain why we don't consider the one odd position after the 6th even one? just like we did in the link I posted in my answer above?
You can delete this questions if you wish. I don't wan to confuse the other people preparing for the exam.[/quote]

We have 6 letters, so only 6 places. There is no 7th place.[/quote]

yes, but when we did the corporation eg, we put a place before and after the entire word (7 places fro 5 vowels). why cant we do the same here?

I'm really weak at this, is there is a gmat club book I could buy instead of annoying everyone (esp Bunuel) here?
Intern
Joined: 21 Aug 2012
Posts: 1
Concentration: Strategy, Finance
In how many different ways can the letters of the word  [#permalink]

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30 Sep 2014, 14:50
1
usre123 wrote:

yes, but when we did the corporation eg, we put a place before and after the entire word (7 places fro 5 vowels). why cant we do the same here?

I'm really weak at this, is there is a gmat club book I could buy instead of annoying everyone (esp Bunuel) here?

The question states that the vowel letters must occupy the odd positions. If you place a vowel in the "7th place" that you are thinking of, then the placement of all letters will shift by one and every vowel will now be in an even position.
Intern
Joined: 08 May 2017
Posts: 1
Re: In how many different ways can the letters of the word  [#permalink]

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29 May 2017, 22:34
I am agree with answer but I have doubt. if option 729 is available additionally than is it possible to consider here repeatation?
3*3*3*3*3*3=729 . three vowels can be arranged on first place ..3 consonants on second ..again 3 vowels on third place..and so in.

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Re: In how many different ways can the letters of the word  [#permalink]

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01 Jun 2017, 09:23
usre123 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

Since detail has 6 letters, there are 3 odd positions, the 1st, 3rd, and 5th spots. Let’s determine how many ways the word can be arranged when the vowels occupy the odd positions.

1st spot: 3 options (any of the 3 vowels)

2nd spot: 3 options (any of the 3 consonants)

3rd spot: 2 options (any of the 2 remaining vowels)

4th spot: 2 options (any of the 2 remaining consonants)

5th spot: 1 option (the last remaining vowel)

6th spot: 1 option (the last remaining consonant)

So, the word can be arranged in 3 x 3 x 2 x 2 x 1 x 1 = 36 ways.

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Re: In how many different ways can the letters of the word  [#permalink]

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07 Dec 2017, 07:32
Top Contributor
usre123 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. So, be sure to learn it.

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Re: In how many different ways can the letters of the word &nbs [#permalink] 07 Dec 2017, 07:32
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