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Intern
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Location: Vietnam
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.
Math Expert
Joined: 02 Sep 2009
Posts: 86800
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|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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Posts: 371
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: 371
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: 154
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 getting
4: 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 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
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 = 120
total 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 = 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
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 6
4. 8! - [6!x6]
5. 5! x [4C3]
Math Expert
Joined: 02 Sep 2009
Posts: 86800
Re: In how many ways can the letters of the word "COMPUTER" be arranged? [#permalink]
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.
_________________
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 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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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 = 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.
_________________
Hope this helps.
You'll never know unless you try
VP
Joined: 11 Aug 2020
Posts: 1407
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