A group consists of four couples, that is, four men each with his own wife. In how many different ways can they be arranged in a straight line such that men and women occupy alternate positions?

Introduction / Context:
This arrangement problem involves both gender and positional constraints. We have four men and four women, each man married to one of the women. They must be lined up in such a way that men and women alternate. The key challenge is to count the number of valid orderings that satisfy the alternating pattern, taking into account that the starting position can be either a man or a woman.

Given Data / Assumptions:

• Total men = 4.
• Total women = 4.
• Each man has exactly one wife, but the constraint is only about gender, not keeping couples together.
• They are to stand in a straight line of 8 positions.
• Men and women must alternate in the line.

Concept / Approach:
An alternating line of 4 men and 4 women can start with either a man or a woman. So there are two possible gender patterns: M W M W M W M W or W M W M W M W M. For each pattern, we first arrange the men in the positions marked for men and then arrange the women in the positions marked for women. Because all individuals are distinct, we use permutations for each gender group and then multiply, finally accounting for the two possible starting patterns.

Step-by-Step Solution:
Step 1: Count arrangements if the line starts with a man. For pattern M W M W M W M W, there are 4 positions for men and 4 positions for women. Number of ways to arrange the 4 men = 4! = 24. Number of ways to arrange the 4 women = 4! = 24. Total arrangements for this pattern = 4! * 4! = 24 * 24 = 576. Step 2: Count arrangements if the line starts with a woman. The pattern W M W M W M W M also has 4 positions for women and 4 positions for men. Arrangements again = 4! * 4! = 576. Step 3: Add the contributions of both patterns. Total valid arrangements = 576 + 576 = 1152.

Verification / Alternative check:
We can factor the calculation as follows: there are 2 choices for whether the first position is occupied by a man or a woman. Once that is fixed, the 4 positions of a given gender can be filled in 4! ways and the other gender positions can be filled in another 4! ways. Hence the total equals 2 * 4! * 4! = 2 * 24 * 24 = 1152. This is a concise confirmation of the result.

Why Other Options Are Wrong:
The values 1278 and 1296 do not correspond to any simple product of factorials under the alternating constraint. The value 432 is much smaller and would result if one of the 4! factors or the factor of 2 was omitted. The option labeled “None of these” is incorrect because we have found a valid count of 1152 that matches one of the choices.

Common Pitfalls:
Students sometimes ignore the possibility of starting with a woman and count only one alternating pattern, leading to exactly half the correct answer. Others mistakenly treat the problem as a simple permutation of 8 distinct people without enforcing the alternating gender pattern, which gives 8! and is far too large. Being careful to first fix the gender pattern, then permute within each group, ensures a correct solution.

Final Answer:
They can be arranged in a straight line with men and women alternating in 1152 different ways.