In how many different ways can the letters of the word

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.

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.

Answer: D.

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?

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.

I am agree with answer but I have doubt. if option 729 is available additionally than is it possible to consider here repeatation?
Means why answer is not
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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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.

Answer: D

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. So, be sure to learn it.

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