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Intern Joined: 05 Feb 2014 Posts: 5 In how many ways can the letters of the word MANIFOLD be arr [#permalink]
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Hide Show timer StatisticsIn how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ? A. 14400 AM GETTING TWO DIFFERENT ANSWERS VIA TWO DIFFERENT METHODS....PLEASE ENLIGHTEN!! Originally posted by ratnanideepak on 05 Feb 2014, 23:19. Renamed the topic and edited the question. Math Expert Joined: 02 Sep 2009 Posts: 86772 Re: In how many ways can the letters of the word MANIFOLD be arr
[#permalink] ratnanideepak wrote: In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ? A. 14400 AM GETTING TWO DIFFERENT ANSWERS VIA TWO DIFFERENT METHODS....PLEASE ENLIGHTEN!! We have 3 vowels: AIO and 5 consonants: MNFLD. Consider the following case: *M*N*F*L*D* If we place vowels in any 3 empty slots (*) then all vowels will be separated by at least one consonant: The # of ways to choose 3 empty slots out of 6 for 3 vowels = \(C^3_6=20\); The # of ways to arrange the vowels: 3! (or instead of these two steps we could use \(P^3_6)\); The # of ways to arrange MNFLD = 5!. Total = 20*3!*5! = 14,400. Answer: A. GMAT Club Legend Joined: 11 Sep 2015 Posts: 6801 Location: Canada
Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
ratnanideepak wrote: In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ? A. 14400 Take the task of arranging the 8 letters and break it into stages. Stage 1: Arrange the 5 CONSONANTS (M, N, F, L and D) in a row IMPORTANT: For each arrangement of 5 consonants, there are 6 spaces where the
VOWELS can be placed. Stage 2: Select a space to place the A. Stage 3: Select a space to place the I. Stage 4: Select a space to place the O. By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus arrange all 8 letters) in (120)(6)(5)(4) ways (= 14,400 ways) Answer: A Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it. RELATED VIDEOS _________________ Brent Hanneson – Creator of gmatprepnow.com Intern Joined: 05 Feb 2014 Posts: 5 Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
another way to solve would be to find out the total ways to arrange the 8 letters and deduct those choices where the vowels are always together. so total number of ways to arrange 8 letters would be 8! choices where vovels are always together would be when you treat 3 vowels as one letter and arrange the remaining 5 letters to give 6! ways to write the letters where the vowels are always together. the vowels can be further arranged in 3! ways. so we have please prove this wrong so that the flaw in the logic is detected.
Math Expert Joined: 02 Sep 2009 Posts: 86772 Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
ratnanideepak wrote: In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ? A. 14400 another way to solve would be to find out the total ways to arrange the 8 letters and deduct those choices where the vowels are always together. so total number of ways to arrange 8 letters would be 8! choices where vovels are always together would be when you treat 3 vowels as one letter and arrange the remaining 5 letters to give 6! ways to write the letters where the vowels are always together. the vowels can be further arranged in 3! ways. so we have please prove this wrong so that the flaw in the logic is detected. The point is that {total} - {all three vowels together} does not give the cases where all the vowels are separated: you still get the cases where any two of them are together. For example, {AI}MNFOLD or MAN{IO}FLD ... Hope it's clear. Intern Joined: 05 Feb 2014 Posts: 5 Re: In how many ways can the letters of the word MANIFOLD be arr
[#permalink] so how do you remove any two of them together from this total ?....i just want to arrive at the correct answer by viewing the entire logic by this method....do we have to deduct 7! * 2 further by treating any two vowels as one...if so the answer is till not the same.... Math Expert Joined: 02 Sep 2009 Posts: 86772 Re: In how many ways can
the letters of the word MANIFOLD be arr [#permalink]
ratnanideepak wrote: In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ? A. 14400 so how do you remove any two of them together from this total ?....i just want to arrive at the correct answer by viewing the entire logic by this method....do we have to deduct 7! * 2 further by treating any two vowels as one...if so the answer is till not the same.... *M*N*F*L*D* Exactly two of the vowels are together. Consider the two vowels as one unit: {X, Y} The # of ways to choose which two vowels
out of three are together = \(C^2_3=3\) {Desired} = {Total} - {All 3 together} - {Exactly two together} = 8! - 6!*3! - 3*2!*6*5*5! = 14,400. Hope it's clear. Director Joined: 25 Apr 2012 Posts: 577 Location: India GPA: 3.21 WE:Business Development (Other)
Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
Bunuel wrote: ratnanideepak wrote: In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ? A. 14400 so how do you remove any two of them together from this total ?....i just want to arrive at the correct answer by viewing the entire logic by this method....do we have to deduct 7! * 2 further by treating any two vowels as one...if so the answer is till not the same.... *M*N*F*L*D* Exactly two of the vowels are together. Consider the two vowels as one unit: {X, Y} The # of ways to
choose which two vowels out of three are together = \(C^2_3=3\) {Desired} = {Total} - {All 3 together} - {Exactly two together} = 8! - 6!*3! - 3*2!*6*5*5! = 14,400. Hope it's clear. Hi Bunuel, I have similar method and getting the same answer but having looked at the solution above, my way of working may not correct. Can you please check Total no. of possible outcomes for MANIFOLD are: 8! Taking 3 vowels as one unit we can arrange the 6 words in 6! ways and among themselves the vowels will arrange in 3! ways *M*N*F*L*D*-------> Now the * can be the position of the vowels together so no. of such words will be 6!*3!*6= 25920 So no. of words that can be formed in which no vowels are together : 8!- 25920= 14400 Ans A
Math Expert Joined: 02 Sep 2009 Posts: 86772 Re: In how many ways can the letters of the word MANIFOLD be
arr [#permalink]
WoundedTiger wrote: Bunuel wrote: ratnanideepak wrote: In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ? A. 14400 so how do you remove any two of them together from this total ?....i just want to arrive at the correct answer by viewing the entire logic by this method....do we have to deduct 7! * 2 further by treating any two vowels as one...if so the answer is till not the same.... *M*N*F*L*D* Exactly two of the vowels are together. Consider the two vowels as one unit: {X, Y} The # of ways to choose which two vowels out of three are together = \(C^2_3=3\) {Desired} = {Total} - {All 3 together} - {Exactly two together} = 8! - 6!*3! - 3*2!*6*5*5! = 14,400. Hope it's clear. Hi Bunuel, I have similar method and getting the same answer but having looked at the solution above, my way of working may not correct. Can you please check Total no. of possible outcomes for MANIFOLD are: 8! Taking 3 vowels as one unit we can arrange the 6 words in 6! ways and among themselves the vowels will arrange in 3! ways *M*N*F*L*D*-------> Now the * can be the position of the vowels together so no. of such words will be 6!*3!*6= 25920 So no. of words that can be formed in which no vowels are together : 8!- 25920= 14400 Ans A There are two problems with your solution: 1. The same as ratnanideepak made in his approach: in-how-many-ways-can-the-letters-of-the-word-manifold-be-arr-167127.html#p1328143 2. The number of arrangements of six units {AIO }{M}{N}{F}{L}{D} is indeed 6!*3! but you don't need further to multiply this by 6, because 6!*3! already gives all the possible arrangements of {AIO }{M}{N}{F}{L}{D}. Hope it's clear. Intern Joined: 17 Jul 2013 Posts: 44 GPA: 3.74 Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
I have done it like this... Attached a non vowel letter to the right OR left of each vowel and make it one unit. (2 ways) M AN IF OL D Now we have 5 entities. we can arrange them in 5! ways. We have 5 non Vowels and 3 Vowels to attach them to, we can select non-vowels to attach with vowels in 5*4*3 ways. Total ways: 2*5!*5*4*3 = 14400 Board of Directors Joined: 17 Jul 2014 Posts: 2230 Location: United States (IL) Concentration: Finance, Economics GMAT 1: 650 Q49 V30 GPA: 3.92 WE:General Management (Transportation)
Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
i got to the answer choice slightly differently...and I dont know if it's a correct way to solve it... now multiply everything: Senior Manager Joined: 29 Jun 2017 Posts: 348 GPA: 4 WE:Engineering (Transportation)
Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
Answer is A: My method is different, consider the following numbers as blank spaces which i will use to denote case 1) vowel at 1,3 then third can be at 5,6,7,8 = 4 case case 2) when vowel at 2,4 then 3rd can be at 6,7,8 = 3 cases total = 6 cases case 3) when vowel at 3,5 then 3rd can be at 7,8 = 2 case case 4) when vowel at 4,6 then 3rd can be at 8 = 1 cases total cases = 10+6+3+1 = 20 in each case total number of arrangement = 3! for vowel and 5! for consonants = 5!3! for 20 cases = 20 x 5! x 3! = 14400 option A GMAT Club Legend Joined: 18 Aug 2017 Status:You learn more from failure than from success. Posts: 7204 Location: India Concentration: Sustainability, Marketing GPA: 4 WE:Marketing (Energy and Utilities) Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
ratnanideepak wrote: In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ? A. 14400 AM GETTING TWO DIFFERENT ANSWERS VIA TWO DIFFERENT METHODS....PLEASE ENLIGHTEN!! great question ; in hurry i calculated not together later realised questions are separated so 2 vowels can still be together UNC Kenan Flagler Moderator Joined: 18 Jul 2015 Posts: 249 GMAT 1: 530 Q43 V20 WE:Analyst (Consumer Products)
Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
Archit3110 wrote: ratnanideepak wrote: In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ? A. 14400 AM GETTING TWO DIFFERENT ANSWERS VIA TWO DIFFERENT METHODS....PLEASE ENLIGHTEN!! great question ; in hurry i calculated not together later realised questions are separated so 2 vowels can still be together Hi Archit3110, I do not understand the part were we consider that there will be 6 spaces to arrange the 3 vowels after the 5 consonants are arranged. Shouldn't there be only 3 spaces after arranging 5 consonants? IMPORTANT: For each arrangement of 5 consonants, there are 6 spaces where the VOWELS can be placed. Warm Regards, Cheers. Wishing Luck to Every GMAT Aspirant | Press +1 if this post helped you! Interested in data analysis & reporting using R programming? - https://www.youtube.com/watch?v=ZOJHBYhmD2I GMAT Club Legend Joined: 18 Aug 2017 Status:You learn more from failure than from success. Posts: 7204 Location: India Concentration: Sustainability, Marketing GPA: 4 WE:Marketing (Energy and Utilities) Re: In how many ways can the letters of the word MANIFOLD be arr
[#permalink] Pritishd Below if you see we have 6 spaces for vowels and total vowels are 3 so possible arrangement for vowels in 6 spaces:6*5*4.. _M_N_D_L_F_ Hope this helps Pritishd wrote: Archit3110 wrote: ratnanideepak wrote: In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ? A. 14400 AM GETTING TWO DIFFERENT ANSWERS VIA TWO DIFFERENT METHODS....PLEASE ENLIGHTEN!! great question ; in hurry i calculated not together later realised questions are separated so 2 vowels can still be together Hi Archit3110, I do not understand the part were we consider that there will be 6 spaces to arrange the 3 vowels after the 5 consonants are arranged. Shouldn't there be only 3 spaces after arranging 5 consonants? IMPORTANT: For each arrangement of 5 consonants, there are 6 spaces where the VOWELS can be placed. Warm Regards, Posted from my mobile device UNC Kenan Flagler Moderator Joined: 18 Jul 2015 Posts: 249 GMAT 1: 530 Q43 V20 WE:Analyst (Consumer Products)
In how many ways can the letters of the word MANIFOLD be arr
[#permalink] Archit3110 wrote: Pritishd Below if you see we have 6 spaces for vowels and total vowels are 3 so possible arrangement for vowels in 6 spaces:6*5*4.. _M_N_D_L_F_ Hope this helps Hi Archit3110, I do not understand the part were we consider that there will be 6 spaces to arrange the 3 vowels after the 5 consonants are arranged. Shouldn't there be only 3 spaces after arranging 5 consonants? IMPORTANT: For each arrangement of 5 consonants, there are 6 spaces where the VOWELS can be placed. Warm Regards, Hi Archit3110, My doubt still remains. I can see that there are 6 spaces but my question was that how did you arrive at those 6 spaces? Once we arrange the 5 consonants will we not have only 3 spaces left? Is this some kind of a method were we assume an available space before and after every element? Warm Regards, Cheers. Wishing Luck to Every GMAT Aspirant | Press +1 if this post helped you! Interested in data analysis & reporting using R programming? - https://www.youtube.com/watch?v=ZOJHBYhmD2I Manager Joined: 22 Sep 2014 Posts: 132 Location: United States (CA)
Re: In how
many ways can the letters of the word MANIFOLD be arr [#permalink]
Con first : A55 A55 * A63 =14400 Posted from my mobile device GMAT Club Legend Joined: 18 Aug 2017 Status:You learn more from failure than from success. Posts: 7204 Location: India Concentration: Sustainability, Marketing GPA: 4 WE:Marketing (Energy and Utilities) Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
Pritishd Well
this is how arrangement questions are solved we need to make cases as per question and solve considering all possible arrangement... Pritishd wrote: Archit3110 wrote: Pritishd Below if you see we have 6 spaces for vowels and total vowels are 3 so possible arrangement for vowels in 6 spaces:6*5*4.. _M_N_D_L_F_ Hope this helps Hi Archit3110, I do not understand the part were we consider that there will be 6 spaces to arrange the 3 vowels after the 5 consonants are arranged. Shouldn't there be only 3 spaces after arranging 5 consonants? IMPORTANT: For each
arrangement of 5 consonants, there are 6 spaces where the VOWELS can be placed. Warm Regards, Hi Archit3110, My doubt still remains. I can see that there are 6 spaces but my question was that how did you arrive at those 6 spaces? Once we arrange the 5 consonants will we not have only 3 spaces left? Is this some kind of a method were we assume an available space before and after every element? Warm Regards, Posted from my mobile device UNC Kenan Flagler Moderator Joined: 18 Jul 2015 Posts: 249 GMAT 1: 530 Q43 V20 WE:Analyst (Consumer Products)
Re: In how many ways can the letters of the word
MANIFOLD be arr [#permalink]
Archit3110 wrote: Pritishd Well this is how arrangement questions are solved we need to make cases as per question and solve considering all possible arrangement... Pritishd wrote: Archit3110 wrote: Pritishd Below if you see we have 6 spaces for vowels and total vowels are 3 so possible arrangement for vowels in 6 spaces:6*5*4.. _M_N_D_L_F_ Hope this helps Hi Archit3110, I do not understand the part were we consider that there will be 6 spaces to arrange the 3 vowels after the 5 consonants are arranged. Shouldn't there be only 3 spaces after arranging 5 consonants? IMPORTANT: For each arrangement of 5 consonants, there are 6 spaces where the VOWELS can be placed. Warm Regards, Hi Archit3110, My doubt still remains. I can see that there are 6 spaces but my question was that how did you arrive at those 6 spaces? Once we arrange the 5 consonants will we not have only 3 spaces left? Is this some kind of a method were we assume an available space before and after every element? Warm Regards, Posted from my mobile device While I understand the other methods that arrangement questions are solved, it is this type that I am unable to figure out. Let me figure out at my end in that case. Cheers. Wishing Luck to Every GMAT Aspirant | Press +1 if this post helped you! Interested in data analysis & reporting using R programming? - https://www.youtube.com/watch?v=ZOJHBYhmD2I Director Joined: 30 Sep 2017 Posts: 968 GMAT 1: 720 Q49 V40 GPA: 3.8
Re: In how many ways can the letters of the word MANIFOLD be arr
[#permalink] 3 vowels AIO and 5 consonants MNFLD to a generalized case: *M*N*F*L*D*, where * can be either empty or filled by any of the three vowels The number of ways to choose 3 empty slots out of 6 for 3 vowels = 6C3 = 20 The number of ways to arrange the vowels= 3! = 6 The number of ways to arrange MNFLD = 5! = 120 Total = 20*6*120 = 14,400. FINAL ANSWER IS (A) Posted from my mobile device Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink] 25 Jan 2020, 11:51 Moderators: Senior Moderator - Masters Forum 3084 posts How many ways can you arrange the letters in the word LOGARITHM?LOGARITHM if each letter is not used more than once. There would be 985 824 arrangements. a) If any of the six letters can be used.
How many of the arrangements of the letters of the word LOGARITHM begin with a vowel and end with a consonant?∴ n(S) = 9! Let E be the event that word starts with vowel and ends with consonant. There are 3 vowels and 6 consonants in the word LOGARITHM. ∴ The first place can be filled in 3 different ways and the last place can be filled in 6 ways.
How many different words can be formed from LOGARITHMS?Solution : The word, 'LOGARITHMS' contains 10 different letters. <br> Number of 4-letter words formed out of 10 given letters <br> `=""^(10)P_(4)=(10xx9xx8xx7)=5040. ` <br> Hence, the required number of 4-letter words `= 5040.
How many ways word arrange can be arranged in which vowels are together?The number of ways the word TRAINER can be arranged so that the vowels always come together are 360.
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