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!.

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