In how many ways can the letters of the word "COMPUTER" be arranged?

On how many ways can the letters of the word "COMPUTER" be arranged?

1. Without any restrictions:
Since all letters in the word "COMPUTER" are distinct then the # of arrangements is 8!.

2. M must always occur at the third place:
M is fixed at the third place, other 7 distinct letters can be arranged in 7! ways,

3. All the vowels are together:
Consider three vowels as one unit: {OEU}. Thus we'll have total of 6 units: {OEU}{C}{M}{P}{T}{R}, which can be arranged in 6! ways. Three vowels within their unit can be arranged in 3! ways. Total: 6!*3!.

4. All the vowels are never together:
Total minus restriction: 8!-6!*3!.

5. Vowels occupy the even positions (the vowels can occupy only even positions):
C|O|M|P|U|T|E|R
O|E|O|E|O|E|O|E (O and E stand for odd and even positions respectively).

# of arrangements would be \(C^3_4*3!*5!=4!*5!=2880\).

\(C^3_4\) - choosing which 3 even positions out of 4 will be occupied by vowels (there are 4 even positions: 2nd, 4th, 6th and 8th and only 3 vowels);
\(3!\) - # of different arrangements of these vowels on their even positions;
\(5!\) - # of different arrangements of 8-3=5 other letters left.

Hope it helps.