Stirling numbers of the second kind for small numbers of partition

Question

I try to do part b of the question below in a manner similar to part a, but still get stuck. Any hints?

My attempt:

Answer 1

Picture three bins and start throwing your elements in them however you like. There is a specific number of ways to do so. However, that also counts the cases that some bin is left empty! You're going to need to subtract those. Then you should promptly add all the ways that left two bins empty, since you counted each twice at the previous step. Finally, the order of the bins doesn't really matter, does it?

Edit your progress if you need extra help.