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

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- Nov 7th 2006, 11:10 PMfeiyingxOrdering 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 - Nov 8th 2006, 04:11 AMSoroban
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: $\displaystyle 8! \,= \,40,320$ 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: .$\displaystyle 40,320 \times 15,120 \;= \;\boxed{609,638,400 \text{ ways.}}$

- Nov 8th 2006, 08:30 AMfeiyingx
Thanks a bunch for the response. xD It was really helpful.