How many ways can 5 people sit in a row if two of them insist on sitting together?
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This is a question about permutations. A permutation is an arrangement of a certain amount of objects in a specific order. Show
For example, if we have two people, #A# and #B#, we can arrange them in a row of two chairs in two ways #A B# or #B A# Each of these two arrangements is a permutation. Thus, there are only two possible permutations in the example. If we have seven people it gets a bit trickier. We must use the multiplication principle. If we have seven people and we want to know how many ways we can arrange them in a row of seven chairs, using the multiplication principle, we multiply all the options together to get the total number of arrangements. E.g. there are 7 options for the first seat, 6 for the second (because one has been used), 5 for the third, 4 for the fourth, and so on. #7*6*5*4*3*2*1=7! =5040# So there are 5040 ways of arranging seven people in a row of seven chairs. I think the proof of this is best seen by making a tree diagram and noting that the number of branches at the end tells you the number of possible arrangements/permutations. The tree for seven people is way too big so if you want to test this, do it with a smaller number, such as three people (there will be #3*2*1=6# branches/arrangements). I won't go into more detail on how this works, other than to say that it is very important and should be understood if you want to deal with permutations or probability. Now, let's call each person #A, B, C, D, E, F, G# The answer is not 5040 because we have a restriction, two of the people must sit next to each other. We'll call Jane and Joe, #A# and #B#. Group them together and call them one object, #X#. Now we effectively have six objects #X, C, D, E, F, G# where #X=A, B# Using the multiplication principle, we can arrange six objects in 6! ways #6*5*4*3*2*1=6! =720# But we also have to take into consideration the group, #X#, which can also be arranged in multiple ways #AB# or #BA# #2*1=2! =2# To take this into account and get the total number of arrangements, we must use the multiplication principle again. This involves multiplying the number of arrangements of six objects by the number of arrangements of the group, #X# #720*2=1440# This makes sense because we have 720 arrangements where #A# is first, but we can have another 720 arrangements where #B# is first. (If they wouldn't want to sit together the answer would be 5! (120) but since there is a pair, we can count them as one person. So, my guess is that the answer is 4! (24)) The person who wrote this answer thought correctly that we need to consider the two person as if they are one so the answer is 4! , but still those two person can switch places between themselves so the correct answer should be 4! * 2! = 48 GMAT Club Daily PrepThank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.Customized we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice we will pick new questions that match your level based on your Timer History Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.Hello Guest!It appears that you are browsing the GMAT Club forum unregistered! Signing up is free, quick, and confidential. Join 700,000+ members and get the full benefits of GMAT ClubRegistration gives you:
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Math Expert Joined: 02 Sep 2009 Posts: 87704 In how many different ways can five people be seated on a five-seat be
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Hide Show timer StatisticsIn how many different ways can five people be seated on a five-seat bench if two of them must sit next to each other? A. 24 _________________ Intern Joined: 25 Jan 2014 Posts: 27 Location: India Concentration: Marketing, Operations GPA: 3.5 WE:Design (Manufacturing)
Re: In how many different ways can five people be seated on a five-seat be
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5 people can sit in 5! ways. Correct answer is B. "I would risk the fall, just to know how it feels to FLY" Intern Joined: 06 Mar 2015 Posts: 20 Re: In how many different ways can five people be seated on a five-seat be [#permalink]
Bunuel wrote: In how many different ways can five people be seated on a five-seat bench if two of them must sit next to each other? A. 24 Lets consider a and b sit together. Hence we have AB, C, D, E. They can sit in 4! ways. Now A and B in between themselves can sit in 2 ways. Hence 4!*2 = 48 ways Current Student Joined: 18 Oct 2014 Posts: 709 Location: United States GPA: 3.98
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Bunuel wrote: In how many different ways can five people be seated on a five-seat bench if two of them must sit next to each other? A. 24 Suppose of two seated together as one. Hence, total of 4 can sit as 4! and two can sit as 2! Total ways that these five can be seated with given condition= 4!*2!= 48 B is the answer I welcome critical analysis of my post!! That will help me reach 700+ Senior Manager Joined: 21 Mar 2016 Posts: 456
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Consider the two of them as one unit, Senior Manager Joined: 24 Nov 2015 Posts: 425 Location: United States (LA)
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consider the seating arrangement as follows Non-Human User Joined: 09 Sep 2013 Posts: 25291 Re: In how many different ways can five people be seated on a five-seat be [#permalink]
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: In how many different ways can five people be seated on a five-seat be [#permalink] 15 Nov 2020, 03:28 Moderators: Senior Moderator - Masters Forum 3097 posts How many ways can you arrange 5 people in a row if 2 refuses to sit next to each other?The number of arrangements without restriction is 5! =120.
How many ways can 5 people sit in a row?So, arrangements of 5 persons can be done in 5! =120 ways. Q. In how many ways can 3 people be seated in a row containing 7 seats ?
How many ways can 5 people be seated around a circular table if two insist to seat next to each other?In how many different ways can five people be seated at a circular table? So the answer is 24.
How many ways can 5 people sit in a bench?ways. Therefore, total different ways in which 5 people can sit are 2! x 4! = 48.
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