# Ordering people in a line

• November 7th 2006, 11:10 PM
feiyingx
Ordering people in a line
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
• November 8th 2006, 04:11 AM
Soroban
Hello, feiyingx!

Quote:

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: $8! \,= \,40,320$ 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: $9\cdot8\cdot7\cdot6\cdot5 \,= \,15,\!120$ choices.

Therefore, there are: . $40,320 \times 15,120 \;= \;\boxed{609,638,400 \text{ ways.}}$

• November 8th 2006, 08:30 AM
feiyingx
Thanks a bunch for the response. xD It was really helpful.