Counting and Probability Question

By stars-and-bars, there are 15 ways.

Well, how many balls can be placed in each box?

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