SOLUTION: In how many ways can 5 men and 2 women sit in a row if no women sit next to each other?

Question 1013622: In how many ways can 5 men and 2 women sit in a row if no women sit next to each other?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
There are ways to sit people in a row.
However, some of those arrangements will have 2 women together.
The two women could be together starting at the first/leftmost position,
or starting at position number 2,
or starting at position number 3, or 4, or 5, or 6.
That is positions where the 2 women can be included next to each other.
For each of those options there are choices of which woman is furthest to the left,
and when all that is decided, there is still the question of how to arrange the men,
and there are, of course, ways to do that.
In all there are ways in which we could have 5 men and 2 women sit in a row with the women next to each other,
out of the ways that 5 men and 2 women can sit in a row.
So, the ways that 5 men and 2 women can sit in a row with no women sitting next to each other are

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