Rosen excercise

Question

How many ways are there for 10 women and 6 men to stand in a line so that no two men stand to each other

Answer 1

First place 10 women in 10! ways.

- W1 - W2 - W3 - W4 - W5 - W6 - W7 - W8 - W9 - W10 -. 

Now there will be 11 position where you can place a man. Select 6 position among these 11, C(11, 6). Now arrange these men with 6! ways.

So answer: 10! * C(11, 6) * 6!.

Answer 2

• First position the $10$ women in $10$! ways.
• Now, $11$ vacant positions are left to place $6$ men in $P(11,6)$ ways.

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Hence, Total ways so that no two men stand to each other = $10$! . $P(11,6)$ ways .