How many ways are there to put 4 distinguishable balls into 2 distinguishable boxes?
Casework!
1) Put all 4 balls into 1 box - 2 ways
2) Put 3 balls into 1 box, 1 ball into other box - 8 ways
3) Put 2 balls in both boxes - 12 ways
Total = 2 + 8 + 12 = 22.
