How many ways are there for eight men and five women to stand in a line
so that no two women stand next to each other?
This approach seems to be the easiest . . .
Place the eight men in a row. .There are: ways.
Leave a space between the men (and on the ends):
. . _ M _ M _ M _ M _ M _ M _ M _ M _
To place the five women, we have a choice of nine spaces.
. . There are: choices.
Therefore, there are: .