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.
Answer: D