There are 10 seats around a circular table. If 8 men and 2 women have to seated around a circular table, such that no two women have to be separated by at least one man. If P and Q denote the respective number of ways of seating these people around a table when seats are numbered and unnumbered, then P : Q equals

Initially we look at the general case of the seats not numbered.
The total number of cases of arranging 8 men and 2 women, so that women are together,
⇒ 8! ×2!

The number of cases where in the women are not together,
⇒ 9! - (8! × 2!) = Q

Now, when the seats are numbered, it can be considered to a linear arrangement and the number of ways of arranging the group such that no two women are together is,
⇒ 10! - (9! × 2!)

But the arrangements where in the women occupy the first and the tenth chairs are not favorable as when the chairs which are assumed to be arranged in a row are arranged in a circle, the two women would be sitting next to each other.

The number of ways the women can occupy the first and the tenth position,
= 8! × 2!

The value of P = 10! - (9! × 2!) - (8! × 2!)
Thus P : Q = 10 : 1