Final answer:
There are 6! × C(7, 4) × 4! ways for 6 women and 4 men to stand in a line so no two men are adjacent, and 7! × 4! ways for all men to stand next to each other.
Step-by-step explanation:
To solve the problem of arranging 6 women and 4 men in a line, we need to consider both scenarios a and b separately.
Scenario a
For the case where no two men stand next to each other, we can first place the 6 women in a line, which can be done in 6! ways. Now we have 7 spaces (before, between, and after the women) where we can place the men so that none of them are adjacent. We select 4 of these 7 spaces, which can be done in C(7, 4) ways, and then arrange the men in those spaces in 4! ways. So the total number of ways is 6! × C(7, 4) × 4!.
Scenario b
When all men must stand next to each other, we can think of the 4 men as a single unit. This unit can be placed in any of the 7 possible spaces just like in scenario a. Once the unit is placed, we have 7 entities to arrange (6 women and the unit of men), which can be done in 7! ways. The men within their unit can be arranged in 4! ways. Therefore, the total ways in this scenario is 7! × 4!.