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?
Can someone explain the solution step by step?
Thank you. =D
Hello, feiyingx!
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: $\displaystyle 8! \,= \,40,320$ 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: $\displaystyle 9\cdot8\cdot7\cdot6\cdot5 \,= \,15,\!120$ choices.
Therefore, there are: .$\displaystyle 40,320 \times 15,120 \;= \;\boxed{609,638,400 \text{ ways.}}$