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In how many ways can 6 people be seated at a round table if [#permalink]
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Question Stats: Hide Show timer StatisticsIn how many ways can 6 people be seated at a round table if one of those seated cannot sit next to 2 of the other 5?a) 720b) 120c) 108d) 84 e) 48
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Re: In how many ways can 6 people be seated at a round table if [#permalink]
Quote: In how many ways can 6 people A,B,C,D,E,F be seated at a round table if B cannot sit next to A and/or C?a) 720b) 120c) 108d) 84 e) 36 Have Modified the Language to make it clearer 6 people are A, B, C, D, E and Fand B can not sit next to A and CConsidering the Position of B is fixed,We have to make sure that 2 person who sit next to B are out of D, E and Fi.e. No. of ways of choosing the neighbours of B = 3C2 = 3 and the no. of ways the Selected neighbours can arrange at the two position adjacent to B = 2!i.e. The ways the B and The neighbours can be arranged = 3C2 *2! = 3*2 = 6 Now The no. of ways in which Remaining Three Individuals can be arranged on remaining 3 seats = 3!Total Ways of making Six person seated such that B doesn't sit next to A and C = 3C2 *2!*3! = 36 _________________
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Re: In how many ways can 6 people be seated at a round table if [#permalink] n people can be seated around a round table in (n-1)! ways. ok...let's find the no of ways in which that person is always seated next to 2 particular people.these 3 can be seated in 2 ways because the cetre position is fixed. now we have a total of 3+1 people...note that 1 represents the group of those 3 people. so 4 can be seated in (4-1)! ways = 6 ways. hence total ways when 2 particular people are always next to one particular of them = 2*6=12 ways.. and total no of ways in which 6 people can be seated =(6-1)!=120 ways.. hence answer= 120=12 = 108 ways. choice b as per me. what's the OA? londonluddite wrote: In how many ways can 6 people be seated at a round table if one of those seated cannot sit next to 2 of the other 5?a) 720b) 120c) 108d) 84 e) 48
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Re: In how many ways can 6 people be seated at a round table if [#permalink] 6 People in a round table can be seated in (6 - 1) ! ways = 120. Now we need to subtract the number of cases when one of those is sitting next to 2 of the other 5. We can consider as if 5 people are sitting in a row because it is round table. Again consider 3 people, those who can not sit together, as a single unit – So the possible arrangements among remaining people 5 – 3 + 1 Unit are = 3 ! And the 3 people unit can arrange among themselves in 3 ! ways. So the possible cases when one of those is sitting next to 2 of the other 5 = 3 ! * 3 ! = 36 Total possible cases = 120 -36 = 84
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Re: In how many ways can 6 people be seated at a round table if [#permalink] OA is C. OE: 6 people can be seated round around a table in 5! ways (would appreciate someones clarification on whether this is correct and why). There are 2 ways the two unwelcome guests could sit next to the person in question and 3! ways of arranging the other three. This is subtracted from 5! giving a result of 108. Clear as mud Edit : AK why can n people be seated in (n-1)! ways and not n! Thanks
Math Expert Joined: 02 Sep 2009 Posts: 87321
Re: In how many ways can 6 people be seated at a round table if [#permalink]
londonluddite wrote: In how many ways can 6 people be seated at a round table if one of those seated cannot sit next to 2 of the other 5?a) 720b) 120c) 108d) 84 e) 48 Check other Seating Arrangements in a Row and around a Table Questions. _________________
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In how many ways can 6 people be seated at a round table if [#permalink] Hi,Can i solve this by taking total number of arrangements i.e 5!=120 ways and then subtract the restriction?120-(arrangements B should not sit next to A and/or C) ?is that a correct approach? GMATinsight wrote: Quote: In how many ways can 6 people A,B,C,D,E,F be seated at a round table if B cannot sit next to A and/or C?a) 720b) 120c) 108d) 84 e) 36 Have Modified the Language to make it clearer 6 people are A, B, C, D, E and Fand B can not sit next to A and CConsidering the Position of B is fixed,We have to make sure that 2 person who sit next to B are out of D, E and Fi.e. No. of ways of choosing the neighbours of B = 3C2 = 3 and the no. of ways the Selected neighbours can arrange at the two position adjacent to B = 2!i.e. The ways the B and The neighbours can be arranged = 3C2 *2! = 3*2 = 6 Now The no. of ways in which Remaining Three Individuals can be arranged on remaining 3 seats = 3!Total Ways of making Six person seated such that B doesn't sit next to A and C = 3C2 *2!*3! = 36
GMAT Club Legend Joined: 08 Jul 2010 Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator Posts: 5920 Location: India GMAT: QUANT EXPERT Schools: IIM (A), ISB WE:Education (Education)
Re: In how many ways can 6 people be seated at a round table if [#permalink]
sunita123 wrote: Hi,Can i solve this by taking total number of arrangements i.e 5!=120 ways and then subtract the restriction?120-(arrangements B should not sit next to A and/or C) ?is that a correct approach? GMATinsight wrote: Quote: In how many ways can 6 people A,B,C,D,E,F be seated at a round table if B cannot sit next to A and/or C?a) 720b) 120c) 108d) 84 e) 36 Have Modified the Language to make it clearer 6 people are A, B, C, D, E and Fand B can not sit next to A and CConsidering the Position of B is fixed,We have to make sure that 2 person who sit next to B are out of D, E and Fi.e. No. of ways of choosing the neighbours of B = 3C2 = 3 and the no. of ways the Selected neighbours can arrange at the two position adjacent to B = 2!i.e. The ways the B and The neighbours can be arranged = 3C2 *2! = 3*2 = 6 Now The no. of ways in which Remaining Three Individuals can be arranged on remaining 3 seats = 3!Total Ways of making Six person seated such that B doesn't sit next to A and C = 3C2 *2!*3! = 36 That would be fine but a difficult approach as you will have to calculate three cases Case-1: When A sits next to B and C does not sit next to B A can sit next to B in 2 ways (On B's right or B's left side)The next adjacent place of B can be occupied in 3 ways because C can't sit next to B Remaining three can sit in 3! waysSo total ways = 2*3*3! = 36 waysCase-2: When C sits next to B and A does not sit next to B C can sit next to B in 2 ways (On B's right or B's left side)The next adjacent place of B can be occupied in 3 ways because A can't sit next to B Remaining three can sit in 3! waysSo total ways = 2*3*3! = 36 waysCase-3: When A and C both sit on either sides of B A and C can sit in 2! ways on two places adjacent to BRemaining three can sit in 3! waysSo total ways = 2!*3! = 12 waysTotal Unfavourable cases = 36+36+12 = 84 waysTotal favourable Cases = (6-1)! - 84 = 120 - 84 = 36 waysI hope it helps! _________________
GMATinsight SUCCESS STORIES: From 620 to 760 l Q-42 to Q-49 in 40 days l 590 to 710 + Wharton l GMAT730+ESCP FREE Resources: 22 FULL LENGTH TESTS l OG QUANT 50 Qn+VIDEO Sol. l NEW:QUANT REVISION Topicwise Quiz
Intern Joined: 21 Jun 2014 Posts: 21
Re: In how many ways can 6 people be seated at a round table if [#permalink] 6 People Sitting Around a Round Table Without Any Restriction = (6-1)! = 5! = 120Restriction = 1 person cannot sit around other two particular peopleComplement Condition = 3 People Will Always Sit togetherNow considering 3 People as one group along with other 3 people , total number of ways they can sit = (4-1) = 3! = 6 WaysBut group of 3 Can also Adjust it self in 3! ways = 6 WaysTotal Complement Ways = 6+6 = 12Total Ways = 120 - 12 = 108 Please correct, if this is not the right way of solving the question
Re: In how many ways can 6 people be seated at a round table if [#permalink] I knew the (n-1)! formula which doesn't solve the entire problem here so I shifted gears and came to the following approach. You have 6 spots on the table. Let's imagine to fix the guy who can't sit with all the other ones on one of the spots.(1) As he can't stand 2 out of total of 5 people, we have 3 options for the 2 seats next to him - so 3C2 which equals 3. (2) For the remaining 3 spots we have 3! or 6 options. Multiply (1) and (2) and you get 18 seating arrangements. But bear with me, we have to remember that we fixed the bad guy on just one spot. He can sit on every seat on the table, namely - 6. So multiply the 18 seating arrangements with 6 and you get 108 (C)
Math Expert Joined: 02 Sep 2009 Posts: 87321
In how many ways can 6 people be seated at a round table if [#permalink]
asicconi wrote: So is the correct answer 36? The original post says 108. The question is a bit ambiguous. Here is the solution to why the answer is 108: in-how-many-ways-can-6-people-be-seated-at-a-round-table-if-36750.html#p253783 Check the discussion HERE for more. _________________
Re: In how many ways can 6 people be seated at a round table if [#permalink]
londonluddite wrote: In how many ways can 6 people be seated at a round table if one of those seated cannot sit next to 2 of the other 5?a) 720b) 120c) 108d) 84 e) 48 This kind of language for students who are in a learning phase is a crime against learning. This is so confusing.The question should have stated that 1 person cannot sit between 2 people. Sitting next does not mean that the person is in between. Example of sentences: I sat next to my nephew and niece who are sitting together. I am sitting next to a couple of bushes in the garden which are adjacent to each other. My house is next to the two red painted houses on the street.If the person does not want to sit in between two specific people.Answer = Total number of cases - Number of cases when he is sitting between these 2 peopleTotal = (6-1)! [number of ways of making people sit in a circle] = 120Number of cases when the person is sitting between the 2 specific people. Place the 3 people aside a bind them together, and now we will consider them as one person. Now, we have 3 free people + 1 hypothetical person (which is actually those 3 binded) = Total 4Formula for circular combination (n-1)! = (4-1)! = 3!Now the three which we binded, can also be arranged.The person which has issues with the other 2 will be in between and the other 2 can only be arranged in 2 ways next to him.So the total ways in which the guy with issues will be stuck between the people he does not want to is 3!*2=12.Answer now is 120-12=108. _________________
Intern Joined: 18 Oct 2020 Posts: 2
Re: In how many ways can 6 people be seated at a round table if [#permalink] The total number of ways of arranging 6 people around a circular table = (6-1)! = 5!=120Assuming A can't sit next to B & C.Going with the complement condition, we assume those cases in which A sits next to B & C. Taking these 3 people as one unit, the total number of ways of arranging the remaining 3 people and this 1 unit (= 3+1) around a circular table is 3!. Now B & C can swap their positions around A as well. So, number of ways are= 3!*2=12. Now, subtracting the complement condition from the total number of possible arrangements, we get our required answer, => 120-12=108.
Re: In how many ways can 6 people be seated at a round table if [#permalink] 31 Jul 2021, 05:08 |
How many ways can 12 people be seated around a circular table if one person must not be moved?
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