Impasta

Seeing as we have a Diophantine Equation with mn, m, and n, it's probably a good idea to use Simon's Favorite Factoring Trick. Move all the terms to one side:

\(mn-2m-4n=0\)

Factor out an m from the first two terms:

\(m(n-2)-4n=0\)

I will leave it to you to figure out what to do next. 
Hint: Try to find a way to get another (n-2). You already have an -4n.

I recommend that you use organized casework to find your answer. For instance, find the probability that you chose two black balls on the first draw, that you chose one black ball and one white ball, and that you chose two white balls.

If I can add on to that, you actually don't need to find the side lengths to find the scale factor. 

Although this is a math forum, I highly recommend using the built-in forum provided by the AoPS classes. These will often give you the best responses, help, and advice. 

This is a Think About It! problem with an elegant solution. So, just think about it!

Here's my (probably not correct) take on this problem. Choose 3 offices out of 5 to be the ones that are assigned to people. Now count how many ways there are to order that; that should be your answer.

Or, another way, first person has 5 choices. How many does the second have?