Problem:
Ten chairs are evenly spaced around a round table and numbered clockwise from through . 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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let the women be seated first. The first woman may sit in any of the chairs. Because men and women must alternate, the number of choices for the remaining women is , and . Thus the number of possible seating arrangements for the women is . Without loss of generality, suppose that a woman sits in chair . Then this woman's spouse must sit in chair or chair . If he sits in chair then the women sitting in chairs , and must have their spouses sitting in chairs , and , respectively. If he sits in chair then the women sitting in chairs , and must have their spouses sitting in chairs , and , respectively. So for each possible seating arrangement for the women there are two arrangements for the men. Hence, there are possible seating arrangements.
The problems on this page are the property of the MAA's American Mathematics Competitions