1. How many permutations of the letters ABCDEFG contain the string BCD? 2.How many ways are there for eight men and five women to stand in a

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.   

BCD and AEFG     I am assuming that the BCD must stay together in that order.

The aefg can be in any order so there are  4! permutations there.

Now for each of those permutations BCD can be before or after the first, or before or after the last or in the middle that is 5 different positions.

So the answer is    4!*5

$${\mathtt{4}}{!}{\mathtt{\,\times\,}}{\mathtt{5}} = {\mathtt{120}}$$

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Sorry I did not see the second question yesterday.