from: category_eng

Counting

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 across from his/her spouse. How many seating arrangements are possible?

$mathrm{(A)} 240qquadmathrm{(B)} 360qquadmathrm{(C)} 480qquadmathrm{(D)} 540qquadmathrm{(E)} 720$

1.	Application

2.

For the first man, there are $10$ possible seats. For each subsequent man, there are $4$ , $3$ , $2$ , and $1$ possible seats. After the men are seated, there are only two possible arrangements for the five women. The answer is $10cdot 4cdot 3cdot 2cdot 1cdot 2 = oxed{480}$ .