An employer interviews 9 people for 5 management
positions in a company, and 4 of the 9 people are men. If all 9 are
qualified, in how many ways can the employer fill the management positions if
exactly 1 is
a man?

4 of the 9 people are men, so therefore presumeably, 5 of the 9 are women.

Let's represent the 5 management positions like this: _ _ _ _ _

Let the first slot be a man. There are 4 of the 9 people that can fit here. The next 4 slots are women, so that would be 5 possible women for the next slot, 4 for the next, 3 for the next, and 2. So that is 4*5*4*3*2= 480 possibilities.

The 1 man could have been in any of the five slots though (not necessarily just the first as we calculated above), so the total number of ways this is possible is 480*5 = 2400 ways.