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Circular Permutation

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Circular Permutation

by MakeitHappen » Fri Feb 19, 2010 11:53 am
I am having difficulty with circular permutations! All I know is am blindly following the formula :(
How to solve this question?
Four men and four women are to be seated alternately at a round table. In how many ways can this be done ?

I am stuck!! Help pls. I could figure out how to do the same if this was a linear permutation though. I.e, if they were to be seated in a row instead of a circle.

Thanks!
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by harsh.champ » Fri Feb 19, 2010 11:59 am
MakeitHappen wrote:I am having difficulty with circular permutations! All I know is am blindly following the formula :(
How to solve this question?
Four men and four women are to be seated alternately at a round table. In how many ways can this be done ?

I am stuck!! Help pls. I could figure out how to do the same if this was a linear permutation though. I.e, if they were to be seated in a row instead of a circle.

Thanks!
Always remember that in circular permutations we have to take (n-1)! and not n!

Anyways for the question I would say:-
Let the 4 women be seated in a spacing first .They can be arranged in (4-1)! = 3! ways.
Now,between these 4 women the men can be arranged in 4!(can be written as 4C1 x 3!) ways.[Here 4! since it also matters between which two women the man is sitting with]

So,the answer will be 4! x 3! = 24 x 6 = 144 ways.

Is my answer correct??
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by MakeitHappen » Fri Feb 19, 2010 12:18 pm
Wow that was fast!! :)
Thanks harsh!
Yep, the answer 144 is correct.
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by harsh.champ » Fri Feb 19, 2010 12:33 pm
Anytime makeithappen.

If you want to know in details as to how the circular permutations work,you should check this link:-
https://www.tutors4you.com/circularpermutations.htm

Over here,it is described with explanations considering both clockwise and anticlockwise order.
Hope it helps.
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by shashank.ism » Fri Feb 19, 2010 11:07 pm
i want to add few tips in this problem....a bit theoretical on
see if you don't have circular permutation ...like students have to seat on bench i a row....you don't have boundation on the last and first student.(like if u were said that two girls can't seat one behind the other..as they are talkative.. (surely they are, it is universal) so in this case you don't have boundation on last and first as they are not related).


but if u consider it in a round table . So first and last is not there. if u start with anyone making it first last will be beside him. SO whatever boundation is given in problem applies to that place also ..
so (n-1)! is use in place of n!
I would rather say go by logic in circular permutation and solve some problems . You will really understand the thing and will not require formulae any more...though formulae will be handy...
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by thephoenix » Sat Feb 20, 2010 12:12 am
MakeitHappen wrote:I am having difficulty with circular permutations! All I know is am blindly following the formula :(
How to solve this question?
Four men and four women are to be seated alternately at a round table. In how many ways can this be done ?

I am stuck!! Help pls. I could figure out how to do the same if this was a linear permutation though. I.e, if they were to be seated in a row instead of a circle.

Thanks!
in normal arrangements of n people its n!
however in circular one its (n-1)!

here 4 women can be seated in around table in 3!
now there is 4 places in b/n them hence in 4! ways can be arranged

tot=144
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