# circular arrangement vs. linear arrangement of people at a table

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?

• Why there is 3! for women ? should not it be 4! ? – A---B Mar 16 '17 at 17:04
• @A---B, it's a circular arrangement, that is, if you have $n$ elements you can arrange them in $n - 1$ ways around a circle. – borat Mar 16 '17 at 17:05
• When you seat the first woman, all chairs are the same, thus the first woman doesn't count in the arrangement. When you seat the first man, each remaining chair is different (different neighbor), thus 4 possibilities. – Alain Remillard Mar 16 '17 at 17:32