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 URL
for 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