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?

$240$

$360$

$480$

$540$

$720$

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.

2008 AMC 10B Problem 21

Solution:

Problem 21 in Other Years