How many ways can four men and four women be seated around a circular table alternating man/woman?
If we seat the women first, there'll be $3!$ arrangements. Then we simply fill in the rest of the seats with men. There should be $4!$ ways to seat men. I thought since we are dealing with a circular arrangement there must be $3!$ ways to seat men, but apparently I thought wrong. Why can men be seated in $4!$ ways rather than $3!$? Are we placing them in a line or something like that?