Question

Write the number of ways 7 men and 7 women can sit on round table such that no two women sit together

Answer 1

Note that if we think as independent chain of women and independent chain on men as the fixing position of men we will get 7!X7! But as we know that a position will also included where it both the chain will revert themselves.
For example
TUVWXYZ and UVWXYZT
ABCDEFG and BCDEFGA are equal so the answer will be 7!X6!

Answer 2

If we see that no two women can sit together it means that women should ait in between two men and for which the number of men are (7-1)! which is 6! and women are 7! ways and the answer is 6!×7!