combination problem

Question

7 men and 7 women are to be seated around a circular table. How many ways can they be seated if no two women can sit next to one another?

I don't have the official answer, it was in the 50 tough questions.

sandeeepsharma · Answer

if no two women can sit next to one another. So first arrange women in circular 
total ways for women to arrange circular = (7-1)!= 6!
Now there are 7 places for 7 man. one place between each 2 women.
total ways to arrange these 7 men into 7 position =7!

so total ways 7 men and 7 women are to be seated around a circular table = 7!*6!

pinchharmonic · Answer

same answer i came to. anyone else to confirm?

KarishmaB · Answer

You are right. The answer is 6! * 7! I have discussed this and some other circular arrangement questions on: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/11 ... 3-part-ii/ (It's Question no. 2) and a variant of this question on: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/12 ... -together/ (Question no. 1)

pinchharmonic · Answer

Karishma,

thanks again. I went through your solution and i want to know if I'm conceptually understanding this right. By placing the first woman, and then saying the other women have 6! places to sit, were you basically proving why sitting at a circular table is (n-1)!? seems like if you place the first woman, without saying she has 7 possible spots, you make everyone else relative to her position so we no longer have to worry about creating extra combinatinos that are equivalent if you just turn the table. because ie she is always say 12:00 position.

then when you seat the men, obviously they are relative to the women so there is no fear of creating extra combinations if you do 7!

Because the way i pictured was they had to alternate, similar to how you did. Then i placed the men in alternating spots. Within that requirement, they still had (7-1)! ways to sit, because of the circular problem. But when the women are placed, they don't have the circular problem because now they are relative to something fixed, ie a particular orientation of the men, so therefore you can just use 7! with the women.

sandeeepsharma · Answer

thanks karishma for confirming

combination problem

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