How many ways are there for 5 men and 11 women to stand in a line where there are at least 2 men in a row?

Question

I have a question from a textbook that I could not really understand what's going on. The question goes like this:

How many ways are there for 5 men and 11 women to stand in a line where there are at least 2 men in a row?

I know that it has something to do with permutation but for the life of me I couldn't find the solution.

Answer 1

HINT: It’s probably easiest to count the arrangements that do not have at least $2$ men in a row and subtract from the $16!$ possible arrangements.

• First line up the $11$ women; in how many ways can this be done?

The women determine $12$ possible locations for men: one at each end of the line, and $10$ more between adjacent women. Each of the $5$ men has to go into a different one of these locations.

• In how many ways can we pick $5$ of the $12$ possible locations?
• In how many ways can the $5$ men be arranged in those $5$ locations?

Now put the pieces together.