There are 8 different letters in the word 'COMPUTER' among them three are vowels and 5 are consonants. We want to arrange all the letters at a time such that: a) All the vowels are always together. It means that we cant arrange three vowels separately with 5 consonants. So, we assume 3 vowels as a single vowel with respect to 5 consonants. Thus, there are all together 6 letters and they can be arranged by 6! ways. But 3 vowels can be arranged to each other by 3! ways. Hence, By multiplication principle of counting:- Total required ways = 6! * 3! = 4320 b) Vowels may occupy only an odd position. Clearly, there are 4 odd positions and 3 vowels. Thus, 3 vowels can be arranged in 4 odd positions by P(4,3) ways. After arranging 3 vowels in an odd position, we arrange 5 consonants in the remaining 5 seats by 5! ways. Hence, By multiplication principle of counting:- Total required ways = P(4,3) * 5! = 24*120 = 2880 c) Relative path of vowels and consonants are not changed. It means, we can't arrange vowels and consonants separately but we arrange vowels with vowels and consonants with consonants. Thus, 5 consonants can be arranged by 5! ways, and 3 vowels can be arranged by 3! ways. Hence, By multiplication principle of counting:- Total required ways = 5! * 3! = 720 Thus in this way, the word COMPUTER can be arranged if the above conditions are given.
Page 2
Intern
Joined: 20 Jul 2005
Posts: 21
Location: Vietnam
In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]
00:00
Question Stats:
Hide Show timer Statistics
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.
Math Expert
Joined: 02 Sep 2009
Posts: 86759
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. _________________
Senior Manager
Joined: 13 Mar 2007
Posts: 374
Schools:MIT Sloan
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: 374
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: 26
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
Manager
Joined: 28 Aug 2006
Posts: 68
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
Manager
Joined: 13 May 2011
Posts: 160
WE 1: IT 1 Yr
WE 2: Supply Chain 5 Yrs
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
Manager
Joined: 25 Nov 2011
Posts: 130
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]
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
Intern
Joined: 03 Oct 2009
Posts: 42
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: 130
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: 86759
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. _________________
Hope this helps.
You'll never know unless you try
VP
Joined: 11 Aug 2020
Posts: 1419
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. _________________
"Do not pray for an easy life, pray for the strength to endure a difficult one." - Bruce Lee
VP
Joined: 11 Aug 2020
Posts: 1419
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! _________________
"Do not pray for an easy life, pray for the strength to endure a difficult one." - Bruce Lee
Non-Human User
Joined: 09 Sep 2013
Posts: 24184
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