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

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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]
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25 Jun 2007, 14:33
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1) 8!
2) 7!
3) 6! x 3!
4) 8! - [6! x 3!]
5) 4! x [4x3x2]
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25 Jun 2007, 19:39
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3) All vowels are together -

C-O-M-P-U-T-E-R has 3 vowels - for them to be together, consider them as a single entity say K,

so now we have 6 alphabets (C,M,P,T,R,K) - 6! ways

K comprises of 3 alphates - so K can arrange itself in 3! ways,

hence total 6! x 3!

5) There are 4 even positions to be filled by 3 vowels -so by direct counting 4 x 3 x 2

remaining 5 positions are occupied by the 5 alphabets in 5! ways

=> 5! x 4!
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26 Jun 2007, 09:53
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
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26 Jun 2007, 12:00
4) All the vowels are never together

Does this mean, they are all seperate? If thats the questions, then asnwer is 4! 5P3 ways
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Re: PS - On how many ways can the letters of the word... [#permalink]

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14 Dec 2011, 11:43
Can someone check for 4 and 5: i keep getting
4: 5!*6p3=2400
5. 720*4=2880
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Re: PS - On how many ways can the letters of the word... [#permalink]

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19 Feb 2012, 04:07
2
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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 )
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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.

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

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19 Feb 2012, 19:33
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?
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Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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19 Feb 2012, 21:24
1
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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.

Kudos please, if this is clear
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Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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19 Feb 2012, 22:53
Thanks Aravind! I understand now...
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Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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20 Feb 2012, 00:43
1. 8!
2. 7!
3. 6! x 6
4. 8! - (6!x6)
5. 5! x (4C3)
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Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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20 Feb 2012, 00:46
docabuzar wrote:
1. 8!
2. 7!
3. 6! x 6
4. 8! - (6!x6)
5. 5! x (4C3)

Answer for question #5 is not correct, it should be 5!*4!. Check the solutions above and ask if anything remains unclear.
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Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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20 Feb 2012, 13:40
Bunuel wrote:
docabuzar wrote:
1. 8!
2. 7!
3. 6! x 6
4. 8! - (6!x6)
5. 5! x (4C3)

Answer for question #5 is not correct, it should be 5!*4!. Check the solutions above and ask if anything remains unclear.

Thanks for correction.
I intended to write 5! x (4P3) => 5! x 4!
I m worried, the time pressure never ceases to cause such mistakes!
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Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]

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

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

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10 Sep 2016, 23:46
For the 5th question-

There are 8 available places in the word COMPUTER. 5 Consonants and 3 Vowels.

First choose 4 consonants to be filled in 4 odd positions in 5P4 ways = 120

Then 4 balance alphabets, including the vowels, can be filled in 4 even positions in 4! ways = 24

Total number of ways = 120*24 = 2880.
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Re: On how many ways can the letters of the word "COMPUTER" be a   [#permalink] 10 Sep 2016, 23:46
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