How many ways can the letters of the word mobile be arranged so that consonants always remain together?

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Permutation and combination Error [#permalink]

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

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Permutation and combination Error [#permalink]

Hey Siddharthk24Welcome to GMATClub!To begin with, the total number of arrangments possible are 61 = 720. However, you arenot taking into consideration arrangements such as MOIBEL,LOIBEM..... where vowels orconsonants are not occupying even spots.In order to solve the above problem, the ideal method is when the word is arranged asfollows: ____Vowel____Vowel____Vowel____The 3 vowels can take any of the three positions and there are 3! ways of arranging thevowels. The consonants can be placed in either of the 4 spots around the vowels & thetotal ways of placing the consonants are \(C_3^4 = 4\) ways. Similarly, there are 3! ways ofarranging the consonants.The total possibilities of arranging MOBILE s.t no consonants are next to each other is4*3!*3! = 144 and number of ways that letters of MOBILE st at least 2 consonants canbe arranged is 720 - 144 = 576Hope that helps! _________________

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Re: Permutation and combination Error [#permalink]

Siddharthk24 wrote:

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

Your mistake is that there aren't just even and odd places for the consonants. You could also have an arrangement like this:

That's why your answer came out too high - you should have subtracted these cases as well.

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Re: Permutation and combination Error [#permalink]

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Re: Permutation and combination Error [#permalink]

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Ans.-144 No. of ways to select 2 consonants from 3=3C2 no. of ways to arrange these two consonants= 3C2*2! no. of ways to arrange remaining terms=4!

Hence, total no. of ways to arrange= 3C2*2!*4!=144