To count circular arrangements without applying a special formula:
1. Place someone in the circle.
2. Count the number of ways to arrange the REMAINING people.
Quote:
In how many different ways 6 men and 6 women can be seated around the table so that no 2 women or 2 men are together?
Here, men and women must ALTERNATE.
After one of the 6 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 = 6. (Any of the 6 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:
6*
5*
5*
4*
4*
3*
3*
2*
2*
1*
1 =
6!5!.
_________________
GMAT and GRE Tutor
Over 1800 followers
GMATGuruNY@gmail.com
New York, NY
If you find one of my posts helpful, please take a moment to click on the "Kudos" icon.
Available for tutoring in NYC and long-distance.
For more information, please email me at GMATGuruNY@gmail.com.