Solution below.

Reveal solution by highlighting:

One way to look at this is: For each bin, choose a random ball. First bin, 5 balls to choose from, second bin, 4 balls to choose from, etc. Put the last ball in a random bin. Then take into account, that the bin with 2 balls could have happened in 2 different ways, depending on which ball was chosen first. We end up with 5*4*3*2 * 4 * 1/2 = 240, the solution.