How many ways can the letters in the word computer be arranged if the letter co must remain next to each other?

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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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In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

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In 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]

Originally posted by sonfbm on 25 Jun 2007, 13:04.
Last edited by Bunuel on 26 Feb 2019, 03:45, edited 1 time in total.

Edited the question.

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

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|RO|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: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

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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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

1) 8! 2) 7! 3) 6! x 3! 4) 8! - [6! x 3!]

5) 4! x [4x3x2]

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

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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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

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: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

Can someone check for 4 and 5: i keep getting4: 5!*6p3=2400

5. 720*4=2880

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

BDSunDevil wrote:

Can someone check for 4 and 5: i keep getting4: 5!*6p3=2400

5. 720*4=2880

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

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

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

What am i doing wrong?

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

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 = 120total 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

Kudos please, if this is clear

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

1. 8!2. 7!3. 6! x 64. 8! - (6!x6)

5. 5! x (4C3)

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

docabuzar wrote:

1. 8!2. 7!3. 6! x 64. 8! - (6!x6)

5. 5! x (4C3)

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

Bunuel wrote:

docabuzar wrote:

1. 8!2. 7!3. 6! x 64. 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.

I m worried, the time pressure never ceases to cause such mistakes!

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

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 = 120Then 4 balance alphabets, including the vowels, can be filled in 4 even positions in 4! ways = 24Total number of ways = 120*24 = 2880. _________________

Hope this helps.

You'll never know unless you try

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In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

COMPUTER is 8 distinct letters.a) No restrictions: 8! / (8 - 8)! = 8! b) M is in the third position: 7! / (7 - 7)! = 7!c) Vowels always together = 6! ( 6-6)! = 6! x 3!d) Vowels never together = 8! - 6! x 3!e) Vowels in even positions:4C3 = 4 < --- # of ways 3 spots can be chosen (Combination)3!/(3-3)! = 6 <--- # of ways 3 vowels can be arranged (Permutation)5!/(5-5)! = 120 <--- # of ways the remaining 5 letters are arranged (Permutation)120 x 24 = 2880 <--- # of ways you can get vowels in even positions. _________________

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

In how many ways can the letters of the word "COMPUTER" be arranged?1) Without any restrictions: 8! 2) M must always occur at the third place: 7! 3) All the vowels are together: 6! x 3! 4) All the vowels are never together: 8! ( 6! x 3!) 5) Vowels occupy the even positions: 5! x 3! _________________

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]

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Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]