The number of ways in which 6 men and 5 women | vedclass

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Question

(A) To arrange $6$ men and $5$ women at a round table such that no two women sit together,we use the gap method.First,arrange the $6$ men at the round

Vedclass · Accepted Answer

$6! 	imes 5!$

Vedclass · Answer

(A) To arrange $6$ men and $5$ women at a round table such that no two women sit together,we use the gap method.First,arrange the $6$ men at the round table. The number of ways to arrange $n$ distinct objects in a circle is $(n-1)!$.So,the number of ways to arrange $6$ men is $(6-1)! = 5!$.This arrangement creates $6$ gaps between the men (including the gap between the last and first man).Since there are $5$ women and they must not sit together,we need to place them in these $6$ available gaps.The number of ways to choose $5$ gaps out of $6$ is $^6C_5$,and the number of ways to arrange the $5$ women in these gaps is $5!$.Thus,the total number of ways is $5! 	imes ^6C_5 	imes 5! = 5! 	imes 6 	imes 5! = 6! 	imes 5!$.

The number of ways in which $6$ men and $5$ women can dine at a round table if no two women are to sit together is given by

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