In how many ways the letters of the word COMPUTER can be rearranged

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

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Distinguishable Ways to Arrange the Word COMPUTER
The below step by step work generated by the word permutations calculator shows how to find how many different ways can the letters of the word COMPUTER be arranged.

Objective: Find how many distinguishable ways are there to order the letters in the word COMPUTER.

Step by step workout:

Input parameters and values:

n1(C) = 1, n2(O) = 1, n3(M) = 1, n4(P) = 1, n5(U) = 1, n6(T) = 1, n7(E) = 1, n8(R) = 1

step 2 Apply the values extracted from the word COMPUTER in the (nPr) permutations equation

= 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8/{(1) (1) (1) (1) (1) (1) (1) (1)}

= 40320/1

Apart from the word COMPUTER, you may try different words with various lengths with or without repetition of letters to observe how it affects the nPr word permutation calculation to find how many ways the letters in the given word can be arranged.

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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]