The Answer to the Question
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There will be 9 places for the 5 women to occupy. They can do it in 9C5 ways.
Moreover, they can arrange themselves in 5! ways.
Also, the men can be arranged in 8! ways.
So, total no of ways = 9C5*8!*5! = 609638400 ways.
How many possible ways are there to arrange eight men in a row? It will be
Now as no two women stand next to each other, we can imagine the situation as
* M * M * M * M * M * M * M * M *
Hence, we need to find how many ways we can arrange 5
5 women in the 9
9 possible (as shown above) places,this is actually
Now applying the fundamental law of counting (precisely product rule), total number of possible arrangements satisfying both constraints is: 15120∗40320=609638400
15120∗40320=609638400 which is your required/desired answer.