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Doubts in Circular permutation.

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Doubts in Circular permutation.

by Dale Steyn » Wed Aug 21, 2013 3:38 am
Hi all. So here is the first question I saw.
1. In how many ways can 5 gents and 5 ladies may sit together in a round table so that there are no two ladies together?

Ans: 5 gents may be arranged by (n-1)! = (5-1)! = 4!
5 ladies occupy remaining positions in 5! way. Therefore we get 4!*5! = 2880
Here is the second question...

2. Find number of ways in the five different flowers can be strung to form a garland?
Lotus,Lily,Jasmine, Red rose and White rose.
The condition is the white rose and red rose will be always together.

My ans: Just like previous problem, I arranged Lotus, Lily and Jasmine as (n-1)! = 2!
Remaining red rose and white rose to be together 2!. The ans as 2!*2! = 4.

 But the given answer is 3!*2! = 12.

I've attached my possible circular permutation for Lotus, Lily and Jasmine [/b]
Attachments
circular permutation.png
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by Mike@Magoosh » Wed Aug 21, 2013 1:29 pm
Dear Dale,
I'm happy to help with this. :-)

In general, the number of ways to permutate (i.e. arrange the order) of n things is n! See this blog:
https://magoosh.com/gmat/2012/gmat-permu ... binations/

For the folks sitting in a circle, it's tricky because the circle doesn't have a beginning or an end, so we arbitrary pick one element and fix that one person, then arrange everyone else (the other n - 1) in sequence from that one fixed pole -- (n-1)!. This assumes, of course, we are concerned only with arrangements vis-a-vis other people at the table, and we are not concerned, say, with how folks are situated with respect to the room in which we find the table (i.e. who can see out of what window, who faces the door, etc.)

You treated the garland as another circle problem, and that's a very sophisticated way to think about it, but I think the question writer was thinking of a garland having a clear break, a clasp or something where it joins, and that this constitutes an asymmetry, such that we couldn't just rotate the same arrangement of flowers around the garland they way that we could rotate the people around the round table. When we extend the string for the garland in a straight line, as we would when we put flowers on it, then it is not a circle --- it has a clear beginning & end. I suspect the question writer was thinking of it in this form.

I agree, your interpretation is not only valid, but actually mathematically insightful, and because of that, I think there's a problem with the question. Rest assured, my friend, the problem does not lie in your mathematical analysis, but in the semantics of question formulation.

Does this make sense?
Mike :-)
Magoosh GMAT Instructor
https://gmat.magoosh.com/
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by Dale Steyn » Thu Aug 22, 2013 10:44 pm
Hey Mike... Thanks man. You really made me feel comfortable now :D B-)
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by Resp007 » Sun Sep 01, 2013 6:07 am
umm, okay how about this explanation?


2. roses needs to be together, so 2!, and now as a one entity. Now we are left with 4 objects in a circular arrangement = (4-1)! = 3!, so answer is = 3! X 2!...
Does this make sense?

1. same logic in 1 is too difficult to apply as a man and a woman can be picked from 5 each = x. then they are 2! and then they are (5-1)!, still something is left to consider it seems :/
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