To find out the number of ways to seat an equal number of men and women alternately at a circular table, calculate (n-1)! arrangements for the men and n! for the women. Since circular arrangements are invariant under rotation, divide by n, resulting in a total of (n-1)! x n!. For 4 men and 4 women, this results in 144 arrangements. The correct answer is option b.
The question concerns the number of ways people can be seated around a circular table such that men and women occupy alternate positions. This is a combinatorial problem often solved using permutations and combinations. Assuming equal numbers of men and women, there are n! ways to arrange n men and (n-1)! ways to arrange n women around a circular table in alternate positions. However, because a circular arrangement is considered the same when rotated, we divide the result by n to correct for overcounting. Assuming n is the common number of men and women:
- Arrange the men in (n-1)! ways (since rotating the circle does not create a new arrangement).
- For each arrangement of men, there are n! ways to insert women into alternate positions.
- Multiply the arrangements of men and women to find the total, which is (n-1)! x n!.
If there are 4 men and 4 women, the calculation would yield 3! (men) x 4! (women) = 6 x 24 = 144 possible arrangements.
Therefore, the correct answer is (b) 144.