There are 7 letters in the word MACHINE
There are 3 vowels: A, I and E
The odd positions are the 1st, 3rd, 5th and 7th positions.

We’ll begin with the most restrictive stage.

Stage 1: Select a position to place the A
There are 4 ODD positions available (the 1st, 3rd, 5th and 7th positions)
So, we can complete stage 1 in 4 ways

Stage 2: Select a position to place the I
After Stage 1 is completed, there are 3 ODD positions remaining.
So, we can complete stage 2 in 3 ways

Stage 3: Select a position to place the E
After Stages 1 and 2 are completed, there are 2 ODD positions remaining.
So, we can complete stage 3 in 2 ways

Stage 4: Select a position to place the M
There are now 4 spaces remaining. So, we can complete stage 4 in 4 ways

Stage 5: Select a position to place the C
There are now 3 spaces remaining. So, we can complete stage 5 in 3 ways

Stage 6: Select a position to place the H
There are now 2 spaces remaining. So, we can complete stage 6 in 2 ways

Stage 7: Select a position to place the N
There is now 1 space remaining. So, we can complete stage 7 in 1 way

By the Fundamental Counting Principle (FCP), we can complete all 7 stages (and thus place al 7 letters) in (4)(3)(2)(4)(3)(2)(1) ways (= 576 ways)

Answer: 576

Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.

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