Question: How many ways can $\mathbf{4}$ men and $\mathbf{4}$ women line up with all the women together and all the men together?
My thoughts: I begin my solution to the problem by adding the total amount of men and women together: $\mathbf{8}$. With that in mind, I find how many ways the men can be lined up. The men can be lined up in $\mathbf{4!}$ ways, likewise the women can be lined up in $\mathbf{4!}$ ways. Thus, the product of these two groups lining up is $\mathbf{576}$.
This is where I can't really understand how the solution $\mathbf{1152}$ is met. Multiplying $\mathbf{576}$ $\cdot$ $\mathbf{2}$ gives me this value, but I don't understand why. What are your thoughts?
Best Answer
The men can go first, or the women can go first. Two options means multiply by two.
For another example, what if there is one man and one woman? How many ways can they line up? Well $1!\times 1!\times 2!$