Question

Write the number of ways in which iin which 6 men and 5 women can dine at a round table if no two women sit together.

Answer 1

If all 11 persons are to be seated in a round table of 11 positions there are exactly 5 places for women and 6 places for men.

It is given that no two women sit together, $5p_5$ this can be done in ways.

The remaining 6 positions can be filled by the 6 men in $6_{p_6}$ ways. so, the fundamental principle of couting, the number of seating arrangements as required is $5_{p_5}\times 6_{p_6}=5!\times 6!$

Hence, the number of ways in which 6 men and 5 women can dine at a round table are $5!\times 6!$.