Description for Correct answer: There are 8 letters in the word 'SOFTWARE including 3 vowels (O, A, E) and 5 consonants (S, F, T, W, R). Considering three vowels as one letter, we have six letters which can be arranged in \( \Large ^{6}P_{6}=6! \) ways. But corresponding to each way of these arrangements, the vowels can be put together in 3! ways. Required number of words = \( \Large 6! \times 3! \) = 4320 Part of solved Permutation and combination questions and answers : >> Aptitude >> Permutation and combination Comments Similar Questions
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In how many ways can the letters of the word MANIFOLD be arr [#permalink]
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Question Stats: Hide Show timer StatisticsIn how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ?A. 14400B. 36000C. 18000D. 24000E. 22200 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.
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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. 14400B. 36000C. 18000D. 24000E. 22200 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. _________________
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. 14400B. 36000C. 18000D. 24000E. 22200 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 We can arrange n unique objects in n! ways. So, we can arrange the 5 consonants in 5! ways (= 120 ways)So, we can complete stage 1 in 120 ways IMPORTANT: For each arrangement of 5 consonants, there are 6 spaces where the VOWELS can be placed. For example, in the arrangement MNDLF, we can add spaces as follows _M_N_D_L_F_So, if we place each vowel in one of the available spaces, we can ENSURE that the vowels are separated.Stage 2: Select a space to place the A. There are 6 spaces to choose from, so we can complete stage 2 in 6 ways. Stage 3: Select a space to place the I. There are 5 remaining spaces to choose from, so we can complete stage 3 in 5 ways. Stage 4: Select a space to place the O. There are 4 remaining spaces to choose from, so we can complete stage 4 in 4 ways. 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: ANote: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it. RELATED VIDEOS_________________
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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 8! - 6!*3!which gives 36000 please prove this wrong so that the flaw in the logic is detected.
Math Expert Joined: 02 Sep 2009 Posts: 86758
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. 14400B. 36000C. 18000D. 24000E. 22200another 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 8! - 6!*3!which gives 36000 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. _________________
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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: 86758
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. 14400B. 36000C. 18000D. 24000E. 22200 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 = \(C^2_3=3\)The # of ways to arrange these two within their unit = 2!;The # of ways to choose an empty slot for that unit = 6;The # of ways to choose an empty slot for the third vowel = 5.The # of ways to arrange MNFLD = 5!.{Desired} = {Total} - {All 3 together} - {Exactly two together} = 8! - 6!*3! - 3*2!*6*5*5! = 14,400.Hope it's clear. _________________
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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. 14400B. 36000C. 18000D. 24000E. 22200 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 = \(C^2_3=3\)The # of ways to arrange these two within their unit = 2!;The # of ways to choose an empty slot for that unit = 6;The # of ways to choose an empty slot for the third vowel = 5.The # of ways to arrange MNFLD = 5!.{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 checkTotal no. of possible outcomes for MANIFOLD are: 8!No. of favourable cases : Total - (Cases in which all 3 vowels are together)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= 25920So no. of words that can be formed in which no vowels are together : 8!- 25920= 14400Ans A _________________
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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. 14400B. 36000C. 18000D. 24000E. 22200 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 = \(C^2_3=3\)The # of ways to arrange these two within their unit = 2!;The # of ways to choose an empty slot for that unit = 6;The # of ways to choose an empty slot for the third vowel = 5.The # of ways to arrange MNFLD = 5!.{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 checkTotal no. of possible outcomes for MANIFOLD are: 8!No. of favourable cases : Total - (Cases in which all 3 vowels are together) 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= 25920So no. of words that can be formed in which no vowels are together : 8!- 25920= 14400Ans 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. _________________
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 DNow 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
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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...we have*C*C*C*C*C*5 consonants, and 6 places for vowels.consonants can be arranged in 5! ways, or 120 ways.1st vowel can be arranged in 6C1 ways2nd vowel can be arranged in 5C1 ways3rd vowel can be arranged in 4C1 waysnow multiply everything: 5!*6C1*5C1*4C1 = 120*6*5*4 = 14,400
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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 1 2 3 4 5 6 7 8case 1) vowel at 1,3 then third can be at 5,6,7,8 = 4 casewhen at 1,4 then = 3 case1,5= 2case1,6 = 1 casetotal = 10 casescase 2) when vowel at 2,4 then 3rd can be at 6,7,8 = 3 caseswhen at 2,5 = 2cases2,6 = 1 casetotal = 6 casescase 3) when vowel at 3,5 then 3rd can be at 7,8 = 2 casewhen at 3,6 = 1 casetotal = 3 casescase 4) when vowel at 4,6 then 3rd can be at 8 = 1 casestotal = 1 casetotal cases = 10+6+3+1 = 20in 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
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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. 14400B. 36000C. 18000D. 24000E. 22200 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 togetherfor consonants ; 5! ways and vowels AIO can be arranged in b/w in 6*5*4 ways total 5! *6*5*4 = 120 * 14400 waysIMO A
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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. 14400B. 36000C. 18000D. 24000E. 22200 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 togetherfor consonants ; 5! ways and vowels AIO can be arranged in b/w in 6*5*4 ways total 5! *6*5*4 = 120 * 14400 waysIMO A 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.
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Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink] PritishdVowels are to be arranged separated.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. 14400B. 36000C. 18000D. 24000E. 22200 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 togetherfor consonants ; 5! ways and vowels AIO can be arranged in b/w in 6*5*4 ways total 5! *6*5*4 = 120 * 14400 waysIMO A 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. Pritishd Posted from my mobile device
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In how many ways can the letters of the word MANIFOLD be arr [#permalink]
Archit3110 wrote: PritishdVowels are to be arranged separated.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. Pritishd 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,Pritishd _________________
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Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink] Con first : A55Vo: 6 slots .. 3 pieces A63A55 * A63 =14400 Posted from my mobile device
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Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink] PritishdWell this is how arrangement questions are solved we need to make cases as per question and solve considering all possible arrangement...You can look for similar questions in GMAT club directory and practice similar type questions. Pritishd wrote: Archit3110 wrote: PritishdVowels are to be arranged separated.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. Pritishd 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,Pritishd Posted from my mobile device
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Re: In how many ways can the letters of the word MANIFOLD be arr [#permalink]
Archit3110 wrote: PritishdWell this is how arrangement questions are solved we need to make cases as per question and solve considering all possible arrangement...You can look for similar questions in GMAT club directory and practice similar type questions. Pritishd wrote: Archit3110 wrote: PritishdVowels are to be arranged separated.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. Pritishd 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,Pritishd 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. _________________
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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 vowelsThe number of ways to choose 3 empty slots out of 6 for 3 vowels = 6C3 = 20The number of ways to arrange the vowels= 3! = 6The number of ways to arrange MNFLD = 5! = 120Total = 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 |
In how many ways can the letters of the word vista be arranged such that the vowels are together
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