Start by counting the number of ways to choose the positions of the vowels: $\binom{7}{3}=35$ Then find how many of these do not have vowels in consecutive positions. With four consonants, there are five places to insert vowels: _c_c_c_c_ Choose three of them: $\binom{5}{3}=10$ So there are $35-10=25$ patterns that meet the condition of having consecutive vowels. For each of these there are $4!$ permutations of the consonants and $3!$ permutations of the vowels. This gives a total of $25\times 4!\times 3!=3600$ permutations with at least two vowels next to each other. For the $10$ patterns that do not have consecutive vowels, there are $4!$ permutations with the vowels in order and $3!$ with the consonants in order. The permutation with both groups in order is counted twice, so there are $4!+3!-1=29$ permutations with at least one group in alphabetical order. This gives a total of $10\times 29=290$ additional permutations that meet the given conditions. The total is therefore $3890$ arrangements.
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In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]
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Question Stats: Hide Show timer StatisticsIn 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. Edited the question.
Math Expert Joined: 02 Sep 2009 Posts: 87762
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!
Senior Manager Joined: 13 Mar 2007 Posts: 364 Schools:MIT Sloan
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]
Intern Joined: 17 Apr 2007 Posts: 25
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 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-ELike 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: 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?
Manager Joined: 25 Nov 2011 Posts: 129 Location: India Concentration: Technology, General Management GPA: 3.95 WE:Information Technology (Computer Software)
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 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
Intern Joined: 17 Jan 2012 Posts: 40
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)
Math Expert Joined: 02 Sep 2009 Posts: 87762
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) Answer for question #5 is not correct, it should be 5!*4!. Check the solutions above and ask if anything remains unclear. _________________
Intern Joined: 17 Jan 2012 Posts: 40
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. 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!
Intern Joined: 03 Aug 2012 Posts: 1 Location: India Concentration: Technology, Healthcare GPA: 3.65 WE:Project Management (Manufacturing)
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. _________________
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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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VP Joined: 11 Aug 2020 Posts: 1388
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] Hello from the GMAT Club BumpBot!Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________
Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink] 23 Feb 2022, 08:54 |
How many words can be formed from the word COMPUTER so that vowels appear in alphabetical order?
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