13. 6 men and 6 women sit around a table, if it is required that men and women alternate, how many possible seating arrangements are there?
13. First, let the 6 gentlemen be seated, then the 6 ladies will be seated between them. There are $(6-1)!=5!$ ways for the 6 gentlemen to be seated, and there are $6!$ ways for the 6 ladies to be seated in between. According to the multiplication principle, the number of seating arrangements is $5! \times 6!$.

Comment: The arrangement of $n$ elements around a closed curve is called a circular permutation, while the usual permutation is a linear permutation. Since a circular permutation has no beginning or end, any element can be rotated around the circle in one direction, which would result in $m$ different methods for a linear permutation, but only one method for a circular permutation. Therefore, the number of circular permutations of $n$ elements is $\frac{n!}{n}=(n-1)!$ kinds.