Counting, permutations and/or combinations

• Mar 26th 2008, 08:30 PM
tony351
Counting, permutations and/or combinations
How many ways can 8 people, 4 men and 4 women, be seated at a round table if people sitting opposite each other must be the opposite gender?

I know this is a circular permutation but I can't seem to come up with this answer.
I thought it would be 7!/2
• Mar 27th 2008, 06:28 AM
awkward
Hi Tony351,

Let's designate one of the men to be Fred and specify the seating relative to Fred's place at the table, numbering the seats counterclockwise as 1,2,3,4,5,6,7,8 with Fred in seat 1.

We can choose the woman opposite Fred in one of 4 ways. There are now 6 people left.

We can choose the person in seat 2 in one of 6 ways and the person opposite in one of 3 ways. There are now 4 people left.

We can choose the person in seat 3 in one of 4 ways and the person opposite in one of 2 ways. There are now 2 people left.

We can choose the person in seat 4 in one of 2 ways and the person opposite in only 1 way. We're done.

Putting it all together, the number of possible arrangements is

4 * (6 * 3) * (4 * 2) * (2 * 1) = 1152.
• Mar 27th 2008, 09:26 AM
iknowone
There are 8 choices for the first person's seat (male or female). Once they sit the opposite chair must be filled by the opposite gender so there are 4 people meeting that criteria, choose 1 to sit.

6 choices for third person (male or female), they sit and there are 3 remaining of the oposite gender one of which must fill the opposite seat. etc.

4 choices for 5th person, 2 choices

2 choices for the 7th person, last one is forced.

8*4*6*3*4*2*2.

The difference here from the proposed answer is the factor of 8 (this counts the 8 rotations of the group around the table for each ordering). You don't count this if the rotation of the table doesn't change the "seating", but the problem isn't entirely clear about that. Since you have the answer as 1152 I suppose they define seating arrangment to be invariant of rotation (true of say a necklace with 4 different black and 4 different white beads) but if you're at dinner I would argue that your actual position (not just your relative position to everyone else at the table) matters. But this is just being nit-picky, the idea is the same.
• Mar 27th 2008, 10:30 AM
tony351
Thank you
Thank you for the detailed answer, cleared up mu confusion :)
• Mar 27th 2008, 05:44 PM
Soroban
Hello, Tony!

Another approach . . .
It's rather primitive, but that's the way my brain works.

Quote:

How many ways can 4 men and 4 women, be seated at a round table
if people sitting opposite each other must be the opposite gender?

Let's seat the men first.

The first man can sit anywhere.

The second man has a choice of the other 7 seats,
. . but must not sit opposite the first man.
Hence, he has 6 choices.

The third man has a choice of the other 6 seats,
. . but he must not sit opposite the first two men.
Hence, he has 4 choices.

The fourth man has a choice of the other 5 seats,
. . but he must not sit opposite the first three men.
Hence, he has 2 choices.

There are: . $6\cdot4\cdot2 \:=\:{\color{blue}48}$ ways for the men to be seated.

The four women can occupy the remaining four seats in $4!= {\color{blue}24}$ ways.

Therefore, there are: . $48 \times 24 \:=\:\boxed{1152}$ seating arrangements.