Thread: Combinatorics

Hello I am in need of urgent help with a problem that involves permutations. The question goes as follows:

The Northeastern University College of Computer and Information Science has 5 women and 20 men who are full-time faculty members. Their photos are to be displayed in a single row at the entrance to their building.

iii. In how many ways can the photos be displayed so that no two women are adjacent? Hint: Consider arranging the men’s photos first and then placing the women’s photos among the men’s photos so as to ensure that no two women are adjacent.

I'm not sure where to start on this one, any help would be greatly appreciated. Thanks!

Hello, checkeredshawn!

The Northeastern University College of Computer and Information Science
has 5 women and 20 men who are full-time faculty members.
Their photos are to be displayed in a single row at the entrance to their building.

(iii) In how many ways can the photos be displayed so that no two women are adjacent?
Hint: Consider arranging the men’s photos first and then placing the women’s photos
among the men’s photos so as to ensure that no two women are adjacent.

That hint tells the whole story . . .

Place the 20 men's photos in a row. .There are ways.

Leave a space between the men's photos and on the ends.
. . _ M _ M _ M _ M _ ... _ M _ M _ M _

To place the women's photos, we choose 5 of the available 21 spaces.
. . There are: . ways.


Therefore, there are: . ways.