2008 AMC 10B Problem 21

Question

Ten chairs are evenly spaced around a round table and numbered clockwise from

1

through

10.

Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or directly across from his or her spouse. How many seating arrangements are possible?

240

360

480

540

720

Answer: C

Concepts:circular arrangements arrangements with restrictions casework

Difficulty rating: 1870

Small hint:

Seat the women first; count their arrangements in the alternating seats

Big hint:

For a fixed seating of the women, count how many ways the men can avoid their spouses

Solution:

Seat the women first. The first woman may take any of the

10

chairs, and since seats alternate, the remaining women fill their four seats in

4!

ways, giving

10\cdot 4!=240

arrangements.

Fix a woman in chair

1.

Her spouse must sit in chair

4

or chair

8;

each choice then forces the placement of every other man consistently. So each seating of the women yields exactly

2

valid seatings of the men.

The total is

2\cdot 240=480.

Thus, the correct answer is C.