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

 Arithmetic is numbers you squeeze from your head to your hand to your pencil to your paper till you get the answer. Carl Sandburg Maths is like a fun & game...! the more you play, the more you enjoy...,, also get knowledge....!! Avika mishra Intern Joined: 02 Dec 2018 Posts: 1 Permutation and combination Error [#permalink]   02 Dec 2018, 21:12 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 Senior PS Moderator Joined: 26 Feb 2016 Posts: 2992 Location: India GPA: 3.12 Permutation and combination Error [#permalink]   02 Dec 2018, 23:09 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! _________________ You've got what it takes, but it will take everything you've gotConnect with me on LinkedIn - https://www.linkedin.com/in/pushpit-chhajer-19388112a/ Manhattan Prep Instructor Joined: 04 Dec 2015 Posts: 941 GMAT 1: 790 Q51 V49 Re: Permutation and combination Error [#permalink]   03 Dec 2018, 13:26 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 doubtYour 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 CThe 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. _________________ Non-Human User Joined: 09 Sep 2013 Posts: 25290 Re: Permutation and combination Error [#permalink]   16 Jul 2021, 08:05 Hello from the GMAT Club BumpBot!Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Re: Permutation and combination Error [#permalink] 16 Jul 2021, 08:05 Moderators: Math Expert 10650 posts Math Expert 87698 posts