If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman, then the number of seating arrangements is

Step 1 Fix the positions of the men.
For arrangements around a circle we may “freeze” one person to remove the rotational symmetry. Fix any one of the $$7$$ men at the top of the table. The remaining $$6$$ men can be permuted in the remaining $$6$$ seats of the circle in $$6!$$ different ways.

Step 2 Identify the gaps for the women.
After all $$7$$ men have been seated, there are exactly $$7$$ empty places—one in between every consecutive pair of men. Because the problem demands a man on either side of every woman, each woman must occupy one of these gaps. No other seats are permissible.

Step 3 Arrange the women in the gaps.
The $$7$$ women are distinct. They can be put into the $$7$$ available gaps in $$7!$$ ways.

Step 4 Multiply the independent choices.
The choices of ordering the men and of ordering the women are independent, so the total number of admissible circular seatings is $$6!\times7!$$

Therefore the required number of seating arrangements is $$6!\,7!$$.
Option A which is: $$6!\,7!$$