In how many different ways

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 ([spoiler]= 36 ways[/spoiler])

Answer: D
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Cheers,
Brent

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