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Director
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The letters of the word PROMISE are arranged so that no two [#permalink]
 22 Mar 2004, 13:50
The letters of the word PROMISE are arranged so that no two
of the vowels should come together. Find total number of arrangements.
1) 49
2) 1440
3) 4320
4) 1898
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Senior Manager
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xPxRxMxSx
4! * 5*4*3 = 1440
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Director
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Hallo guys, I also got 1440 _P_R_M_S_ or 5C3x4!x3!=1440. there must be a mistake coz the official answer is 4320.Thanx!
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Senior Manager
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multiply it with #placing consonants. You will have desired answer
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Director
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Paul think that you have missed the 4! for the consonants so you answer is also 10x3!x4!. Thanx
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GMAT Club Legend
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Let X = vowel
Possible arrangements:
X_X_X_ _
X_X_ _X_
X_X_ _ _X
X_ _X_X_
X_ _X_ _X
X_ _ _X_X
_X_X_X_
_X_X_ _X
_X_ _X_X
_ _X_X_X
In each of above possible ways (10 in total) of placing vowels, you can switch vowels around 3! ways: 3! = 6
Number of ways of arranging consonants: 4! = 24
Since there are 10 ways of arranging vowels: 10(3!*4!) = 1440
Sorry I had to edit my answer because I forgot about consonants 
_________________
Best Regards,
Paul
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GMAT Club Legend
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Wow, quick on correcting mistakes. There was no answer when I started typing and 4 of them popped up by the time I finished! You guys are quick!
_________________
Best Regards,
Paul
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Director
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kpadma wrote: I'm getting 720.
Total ways to arrange = 7!
there are 3 vowels
ways to pick 2 vowels out of 3 is 3C2 = 3
Ways to arrange the picked two vowels = 2
Ways to arrange 2 vowel combination with other 5 letters = 6!
Total ways to arrange 7 letters with 2 vowels togather = 6 * 6!
Total ways to arrange 7 letter with vowels seperated = 6! = 720
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Answer should be 4320
No of ways in which vowels are together = no of ways in which exactly two vowels are together + no of ways in which 3 vowels are together.
But if you just choose 2 vowels in 3C2 and consider it as a set , you will have this set, one remaining vowel and 4 other consonants.
If you rearrange these it also covers set containgin 3 vowels together.
so you have 2 * 3C2 * 5! as invalid combinations
No of ways to arrange 7 letters = 7! = 5040
Desired ways = 5040 - 2 * 3C2 * 5! = 4320
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Senior Manager
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#no of ways in which 3 vowels are together and #ways 2 vowels are together are not mutually exclusive.
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anandnk wrote: Answer should be 4320
No of ways in which vowels are together = no of ways in which exactly two vowels are together + no of ways in which 3 vowels are together.
But if you just choose 2 vowels in 3C2 and consider it as a set , you will have this set, one remaining vowel and 4 other consonants.
If you rearrange these it also covers set containgin 3 vowels together.
so you have 2 * 3C2 * 5! as invalid combinations
No of ways to arrange 7 letters = 7! = 5040
Desired ways = 5040 - 2 * 3C2 * 5! = 4320
Anand,
Just wondering if ur solution takes into account the different ways in which 3 vowels will be together and there placement along with respect to consonants.
cheers
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Senior Manager
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kpadma wrote: kpadma wrote: I'm getting 720. Total ways to arrange = 7! there are 3 vowels ways to pick 2 vowels out of 3 is 3C2 = 3 Ways to arrange the picked two vowels = 2 Ways to arrange 2 vowel combination with other 5 letters = 6! Total ways to arrange 7 letters with 2 vowels togather = 6 * 6! Total ways to arrange 7 letter with vowels seperated = 6! = 720 well, you assumed words such as xxVyVza have no vowles together. For example, x, y or z can be the one Vowel you didn't choose.
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Senior Manager
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anandnk wrote: Answer should be 4320
No of ways in which vowels are together = no of ways in which exactly two vowels are together + no of ways in which 3 vowels are together.
But if you just choose 2 vowels in 3C2 and consider it as a set , you will have this set, one remaining vowel and 4 other consonants.
If you rearrange these it also covers set containgin 3 vowels together.
so you have 2 * 3C2 * 5! as invalid combinations
No of ways to arrange 7 letters = 7! = 5040
Desired ways = 5040 - 2 * 3C2 * 5! = 4320
Your method makes the most sense. Yes 4320 should be the correct answer. Thanks
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word PROMISE has vowels O,I,E
if you chose two vowels then you will have following combinations
OI - it can also also be arranged as IO
IE - it can also be arranged as EI
OE - it can also be arranged as EO
Consider one combination OI
Now you have PRMSE and OI
Following are some of the combinations that can be obtained
ESMRPOI
ESMRPIO
and
SMRPEIO - > Here all the 3 vowels are together
So just accounting for two vowels at a time should cover everything.
Thus for each vowel pair we have invalid combinations as
2 * 5! -> for OI and IO
2 * 5! -> for IE and EI
2 * 5! -> for EO and OE
this is same as 2 * 3C2 * 5! = 720
Total combinations are 7! = 5040
so valid combinations are 7! - 720 = 4320
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Sorry I did a mistake
it should be 2 * 6!
So invalid combinations are 2 * 3C2 * 6! = 4320
Valid combinations are 720 then
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Senior Manager
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anandnk wrote: Sorry I did a mistake
it should be 2 * 6!
So invalid combinations are 2 * 3C2 * 6! = 4320
Valid combinations are 720 then Now, you can see overlapping permutations. Quote: Set A: 2 * 6! -> for OI and IO
Set B: 2 * 6! -> for IE and EI
Set C: 2 * 6! -> for EO and OE
ESMRPOI
ESMRPIO
and
SMRPEIO - > Here all the 3 vowels are together
You will find same permutations in either of the sets A, B, or C--the permutations with 3 vowels together.
For example,
SMRPEIO is counted both in set A(IO/OI) and in set B(EI/IE).
SMRPEOI is counted in set A and in set C and so on.
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