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

1 Answer

Say the men are A,B,C, and D, and the women E,F,G, and H. Because the table is circular, you can indeed place A anywhere, but once A is seated, the 3 remaining men can be seated in 3! ways, since the three remaining seats for the three men are all different with respect to A.

But note, once A is seated, the same holds true for the four women as for the three remaining men: each seat becomes different. That is, as soon as A is seated, then there are 4 different seats for E with respect to A. So, you get 4! Possibilities for the four women.

So, yes, in a way, once you seat 1 person, the circle is 'broken', and it becomes like a linear arrangement after that.

Of course, we could also have seated E first, giving us 3! possibilities for the three remaining women, and 4! for the four men.

Either way, there are 3!4! different seating arrangements.