2.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
ok, here is the line
?M?M?M?M?M?M?M?M?
The M represents a man and the ? represents where a woman could stand
there are 8! permutations for the men (8P8)
Now there are 9 places where the 5 women could stand so that is 9P5
Put them together and you have 8!*9P5 permutations
$${\mathtt{8}}{!}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{9}}{!}}{({\mathtt{9}}{\mathtt{\,-\,}}{\mathtt{5}}){!}}}\right)} = {\mathtt{609\,638\,400}}$$
I think that is correct. 