5 A conference room as m men and w women and m w chairs a T

-5- A conference room as m men and w women and m + w chairs. -a- Two of the men always sit together. Find the probability that all women are adjacent to each other. -b- Find the probability that no two women are adjacent to each other.

Solution

5.

This all condition is for round chair

a. the m men and w women sit randomly, except that two specific men out of the m men always sit together.
With the two men sitting together, you can treat this as m-1 men and w women, and m+w-1 chairs.
There are (m+w-1)! ways to arrange the groups.

If you want the women to sit together, then they all must sit at an end, or between two men. The m-1 male groups can be arranged in (m-1)! ways. there are m places you can put the women. Viewing the chairs left to right, you can put the women to the left of any of the m-1 groups. The women can be arranged in w! ways.

So there are (m-1)! x m x w! =m! w! ways to satisfy the conditions.
The probability is m! w! / (m+w-1)!

b. When no two women are adjacent to each other

You now have (m+w)! possible arrangements. To arrange all the men, in m! ways, and all the women, in w! ways,

With m men, you have m+1 places to put the women. So, putting the women in between the men there is no adjacent when the w<m that is the probability is zero. If w>m then the probability that no two women are adjacent to each other is not zero.