toddmartin wrote:

Can you explain why the ways the men can be seated is 2! and not 3!

Since I cannot attach any figures, refer to

this URLfor explanation of circular permuations. This is my explanation...

1-2-3-4-5-6-1 (assume this to be a closed circle)

Lets assume the position of the first man to be "fixed" at 1, then there are two remaining positions for the remaining two men: 3rd and 5th seat.

Number of ways of seating a man in 3rd seat = 2

As only one man is left, number of ways of seating a man in 5th seat is 1.

Number of ways that three men can be seated first at the round table = 1*2*1=2

There are three remaining seats 2,4, and 6.

Number of options for 2 = 3

Number of options for 4 = 2 (only 2 women are left)

Number of options for 6 = 1 (only 1 woman left)

i.e. The three women can be seated in 3! ways = 6

Total number of possibilities = 2*6 = 12