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On how many ways can the letters of the word "COMPUTER" be a [#permalink]
25 Jun 2007, 13:04
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Question Stats:
83% (01:42) correct
17% (00:00) wrong based on 25 sessions
On 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] _________________
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
Re: PS - On how many ways can the letters of the word... [#permalink]
19 Feb 2012, 03:07
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This post received KUDOS
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 ) _________________
Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]
19 Feb 2012, 03:27
5
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Expert's post
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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.
Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]
19 Feb 2012, 20:24
1
This post received KUDOS
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.
Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]
16 Sep 2013, 21:18
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Re: On how many ways can the letters of the word "COMPUTER" be a [#permalink]
30 Jan 2015, 09:47
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