LUANDATO wrote:Seven men and seven women have to sit around a circular table so that no 2 women are together. In how many ways can that be done?
A. 5!*6!
B. 6!*6!
C. 5!*7!
D. 6!*7!
E. 7!*7!
To count circular arrangements:
1. Place someone at the table.
2. Count the number of ways to arrange the REMAINING people.
Here, men and women must ALTERNATE.
After one of the 7 men has been placed at the table, count the number of options for each empty seat, moving clockwise around the table:
Number of options for the first empty seat = 7. (Any of the 7 women.)
Number of options for the next empty seat = 6. (Any of the 6 remaining men.)
Number of options for the next empty seat = 6. (Any of the 6 remaining women.)
Number of options for the next empty seat = 5. (Any of the 5 remaining men.)
Number of options for the next empty seat = 5. (Any of the 5 remaining women.)
Number of options for the next empty seat = 4. (Any of the 4 remaining men.)
Number of options for the next empty seat = 4. (Any of the 4 remaining women.)
Number of options for the next empty seat = 3. (Any of the 3 remaining men.)
Number of options for the next empty seat = 3. (Any of the 3 remaining women.)
Number of options for the next empty seat = 2. (Either of the 2 remaining men.)
Number of options for the next empty seat = 2. (Either of the 2 remaining women.)
Number of options for the next empty seat = 1. (Only 1 man left.)
Number of options for the last empty seat = 1. (Only 1 woman left.)
To combine these options, we multiply:
7*
6*
6*
5*
5*
4*
4*
3*
3*
2*
2*
1*
1 =
6!7!
The correct answer is
D.
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