How many ways are there to put 4 distinguishable balls into 2 distinguishable boxes?
Well, there are 2 boxes, so I have 2 choices. Then, for the second ball, I also have 2 choices, regardless of where I put the first ball. In fact, for each ball, I have 2 choices, since the boxes are distinguishable. So, the total number of ways to put 4 balls into 2 boxes is 2 times 2 times 2 times 2, or 2^4. That's 16. 