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

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 Correct Answer 10 : 1

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!) = QNow, 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
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