sukh wrote:How many ways can the letters of the word COMPUTER be arranged so that vowels occupy even positions ?

Vowels: O, U, E
Consonants: C, M, P, T, R

We can also use the Fundamental Counting Principle (FCP) to solve this question (in fact most counting questions can be solved using the FCP)

We'll take the task of building words and break it into stages.
Note: There are 8 spaces for the 8 letters.

Stage 1: place a letter in the 1st space
Stage 2: place a letter in the 2nd space
Stage 3: place a letter in the 3rd space
.
.
.
Stage 8: place a letter in the 8th space

We'll first fill the spaces that hold consonants (spaces 1, 3, 5, and 7)
Stage 1: There are 5 consonants to choose from, so this stage can be accomplished in 5 ways.
Stage 3: Once we have placed a consonant in space #1, there are 4 consonants remaining, so this stage can be accomplished in 4 ways.
Stage 5: Can be accomplished in 3 ways.
Stage 7: Can be accomplished in 2 ways.

Now we'll fill the remaining spaces (with 3 vowels and 1 remaining consonant).
Stage 2: There are 4 letters remaining, so this stage can be accomplished in 4 ways.
Stage 4: There are 3 letters remaining, so this stage can be accomplished in 3 ways.
Stage 6: This stage can be accomplished in 2 ways.
Stage 8: This stage can be accomplished in 1 way

So, the total number of ways to complete all 8 stages (and fill all 8 spaces) is:
5x4x3x2x4x3x2x1=2880

Cheers,
Brent