Question

In how many ways can the letters in the word computer'be arranged if co'remain next to each other?

Answer 1

Answer:

Explanation:

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 C34∗3!∗5!=4!∗5!=2880C43∗3!∗5!=4!∗5!=2880.

C34C43 - 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!3! - # of different arrangements of these vowels on their even positions;

5!5! - # of different arrangements of 8-3=5 other letters left.

Hope it helps.  

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