How many ways are there to put 5 balls in 2 boxes if the balls are distinguishable and the boxes are distinguishable?
Assuming a box could be empty
Choose all 5 balls to go into either box - the other box is empty by default
C(5,5) * 2 = 2 ways
Choose 4 of the 5 balls to go into either box - the remaing ball goes into the other box by default
C(5,4) *2 = 10 ways
Choose 3 of the 5 balls to go into either box - the remaining two balls go into the other box by default
C(5,3) * 2 = 20 ways
32 ways