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

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
Option D is the answer

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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.

Answer: D

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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