### Instruction: How many ways are there to put 4 distinguishable balls into 2 indistinguishable boxes?### Response: To start, I'll label the balls A, B, C, and D for convenience.
Then I'll think about how many balls can go in the first box, and how many ways there are to choose them.
The first box can have either 0, 1, 2, 3, or 4 balls.
If it has 0 balls, there is only 1 way to do that: leave all the balls in the second box.
If it has 1 ball, there are 4 ways to do that: choose any one of the balls and put it in the first box, and leave the rest in the second box.
If it has 2 balls, there are 6 ways to do that: choose any two of the balls and put them in the first box, and leave the other two in the second box.
This is the same as the number of combinations of 4 things taken 2 at a time, which is 4 choose 2, or (4 * 3) / (2 * 1) = 6.
If it has 3 balls, there are also 4 ways to do that: choose any three of the balls and put them in the first box, and leave the remaining one in the second box.
This is the same as the number of combinations of 4 things taken 3 at a time, which is 4 choose 3, or (4 * 3 * 2) / (3 * 2 * 1) = 4.
If it has 4 balls, there is only 1 way to do that: put all the balls in the first box and leave the second box empty.
So, adding up all the possibilities, there are 1 + 4 + 6 + 4 + 1 = 16 ways to put 4 distinguishable balls into 2 indistinguishable boxes.