Dblooos wrote:Find the number of ways the letters of the word "MACHINE" can be arranged so that vowels may occupy only odd positions?

A. 3! X 4!
B. 7P3 X 4!
c. 7P4 X 3!
D. None of these

Your solution looks great, Dbloos. Here's my solution with some rationale added.


Take the task of building words and break it into stages.

We have 7 spaces for letters: _ _ _ _ _ _ _

We'll begin with the most RESTRICTIVE stages (placing the vowels)

Stage 1: Place the A
The A can go in spaces #1, #3, #5 or #7, so we can complete stage 1 in 4 ways

Stage 2: Place the I
Now that we've placed the A, there are 3 acceptable spaces remaining, so we complete this stage in 3 ways.

Stage 3: Place the E
Now that we've placed the A and I, there are 2 acceptable spaces remaining, so we complete this stage in 2 ways.

Stage 4: Place the M
There are 4 spaces remaining, so we complete this stage in 4 ways.

Stage 5: Place the C
There are 3 spaces remaining, so we complete this stage in 3 ways.

Stage 6: Place the H
We complete this stage in 2 ways.

Stage 7: Place the N
We complete this stage in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 7 stages (and thus build 7-letter words) in (4)(3)(2)(4)(3)(2)(1) ways ([spoiler] = D[/spoiler])

Cheers,
Brent

Aside: For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775