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On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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25 Jun 2007, 14:04

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

1) Without any restrictions. 2) M must always occur at the third place. 3) All the vowels are together. 4) All the vowels are never together. 5) Vowels occupy the even positions[/b] _________________

Thank you. somehow i can't be sure of my answers, when it comes to arrangement possibilities or probability calculation. Though i got first 4 correct here

Re: PS - On how many ways can the letters of the word... [#permalink]

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19 Feb 2012, 04:07

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BDSunDevil wrote:

Can someone check for 4 and 5: i keep getting 4: 5!*6p3=2400 5. 720*4=2880

4) All the vowels are never together. This is equivalent to (All possibilities - All the vowels are ALWAYS together) = 8! - 6!3!

5) Vowels occupy the even positions. let us consider the following: first 3 vowels placing together in even positions: -O-U-E-- -O---U-E ---O-U-E Like this, at any point in time we have 4 positions to fill with 3 letters. Hence no. of ways will be 4P3 = 4!

Remaining 5 positions can be filled by 5!

Hence total ways = 4! x 5!

Hope this is clear. (if you like, give me kudos, please ) _________________

Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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19 Feb 2012, 04:27

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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.

Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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19 Feb 2012, 21:24

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Apex231 wrote:

4. All the vowels are never together:

I did this -

3 vowels and 5 non-vowels.

number of ways to arrange 5 non-vowels = 5! (represented by | below).

-|-|-|-|-|-

Now there are 6 places (represented by -) that vowels can occupy so that they are not together. Number of ways vowels can be arranged = 6P3 = 120

total number of ways = 120 * 5! = 14400.

What am i doing wrong?

Question says "ALL the vowels not together". So, you have excluded valid cases like COMPTUER, CMPOUTER

One thing: it is generally a good practice to find the probability of something to occur and then subtract it from 1 to find for the same thing to not occur. This way, we don;t commit above mistakes.

Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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16 Sep 2013, 22:18

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Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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