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# In how many ways can 4 men and 4 women sit at a round table so that no

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Re: In how many ways can 4 men and 4 women sit at a round table so that no [#permalink]
4 women can sit around a table in 3! Ways and now there are 4 spaces for men. In which they can sit in 4! Ways.
Total number of ways =3!×4!= 6×24=144

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Re: In how many ways can 4 men and 4 women sit at a round table so that no [#permalink]
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Bunuel wrote:
In how many ways can 4 men and 4 women sit at a round table so that no two men are adjacent? (Two sitting arrangements are considered different only when the positions of the people are different relative to each other.)

(A) 24
(B) 48
(C) 72
(D) 144
(E) 288

In how many ways can 4 men and 4 women sit at a round table with no two women in consecutive postions?

Solution:

Let’s first assume the 8 people were arranged in a line rather than a circle. Then, since no two men are allowed to sit together, the 8 people should be arranged as M - W - M - W - M - W - M - W or W - M - W - M - W - M - W - M. For each of these two scenarios, the number of arrangements is 4 x 4 x 3 x 3 x 2 x 2 x 1 x 1. In total, the number of arrangements where no two men sit together is 2 x 4 x 4 x 3 x 3 x 2 x 2 x 1 x 1.

Now, let’s go from the linear case to the circular case. Since there are 8 people in total, 8 arrangements in the linear case will correspond to one arrangement in the circular case. To see why, let the four men be A, B, C and D; and let the four women be 1, 2, 3 and 4. We see that the arrangements A - 1 - B - 2 - C - 3 - D - 4, 1 - B - 2 - C - 3 - D - 4 - A, B - 2 - C - 3 - D - 4 - A -1 etc. all correspond to the same circular arrangement. Thus, the total number of arrangements around the circular table is (2 x 4 x 4 x 3 x 3 x 2 x 2)/8 = 4 x 3 x 3 x 2 x 2 = 144.

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Re: In how many ways can 4 men and 4 women sit at a round table so that no [#permalink]
Lets seat women first, as there is no marked seat , no of way = 3!
Now there are places and 4 men are to be seated , this can be done in 4! ways .
Ans = 3! * 4!
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Re: In how many ways can 4 men and 4 women sit at a round table so that no [#permalink]
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Re: In how many ways can 4 men and 4 women sit at a round table so that no [#permalink]
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