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02 Dec 2018, 22:12
Kudos
Bookmarks
in how many ways can letter of word MOBILE can be arranged so that at least two consonant remains together?
I was trying to do this way
total no of arrangements - arrangements in which consonants are never together
total no of arrangements - 6!
arrangements in which consonants are never together :
case 1: consonants are at even places then they can be arranged in 3! ways and remaining vowels can be arranged in 3! ways. so total ways = 3! * 3! = 36
case 2: consonants are at odd places then they can be arranged in 3! ways and remaining vowels can be arranged in 3! ways. so total ways = 3! * 3! = 36
so my answer is 720 - 72 = 648
which is apparently wrong. need some help to clear my doubt
I was trying to do this way
total no of arrangements - arrangements in which consonants are never together
total no of arrangements - 6!
arrangements in which consonants are never together :
case 1: consonants are at even places then they can be arranged in 3! ways and remaining vowels can be arranged in 3! ways. so total ways = 3! * 3! = 36
case 2: consonants are at odd places then they can be arranged in 3! ways and remaining vowels can be arranged in 3! ways. so total ways = 3! * 3! = 36
so my answer is 720 - 72 = 648
which is apparently wrong. need some help to clear my doubt
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03 Dec 2018, 00:09
Kudos
Bookmarks
Hey Siddharthk24
Welcome to GMATClub!
To begin with, the total number of arrangments possible are 61 = 720. However, you are
not taking into consideration arrangements such as MOIBEL,LOIBEM..... where vowels or
consonants are not occupying even spots.
In order to solve the above problem, the ideal method is when the word is arranged as
follows: ____Vowel____Vowel____Vowel____
The 3 vowels can take any of the three positions and there are 3! ways of arranging the
vowels. The consonants can be placed in either of the 4 spots around the vowels & the
total ways of placing the consonants are \(C_3^4 = 4\) ways. Similarly, there are 3! ways of
arranging the consonants.
The total possibilities of arranging MOBILE s.t no consonants are next to each other is
4*3!*3! = 144 and number of ways that letters of MOBILE st at least 2 consonants can
be arranged is 720 - 144 = 576
Hope that helps!
Welcome to GMATClub!
To begin with, the total number of arrangments possible are 61 = 720. However, you are
not taking into consideration arrangements such as MOIBEL,LOIBEM..... where vowels or
consonants are not occupying even spots.
In order to solve the above problem, the ideal method is when the word is arranged as
follows: ____Vowel____Vowel____Vowel____
The 3 vowels can take any of the three positions and there are 3! ways of arranging the
vowels. The consonants can be placed in either of the 4 spots around the vowels & the
total ways of placing the consonants are \(C_3^4 = 4\) ways. Similarly, there are 3! ways of
arranging the consonants.
The total possibilities of arranging MOBILE s.t no consonants are next to each other is
4*3!*3! = 144 and number of ways that letters of MOBILE st at least 2 consonants can
be arranged is 720 - 144 = 576
Hope that helps!
03 Dec 2018, 14:26
Kudos
Bookmarks
Siddharthk24
Your mistake is that there aren't just even and odd places for the consonants. You could also have an arrangement like this:
C V V C V C
The first consonant is in an odd place, and the other two are in even places. But, the consonants aren't together.
That's why your answer came out too high - you should have subtracted these cases as well.
