## Online Course Discussion Forum

### I don't understand the solution

The question is:

How many ways are there to put $10$ numbered balls into $4$ numbered boxes (a box is allowed to be empty)?

The solution says:

Each ball can be in any of the $10$ boxes and they are not identical balls, so there are $4^{10} = 1048576$ ways of putting the $10$ balls in the $4$ boxes.

I am unable to understand that why the answer is 4 to the 10th power. On the other hand, I think the condition that says "a box is allowed to be empty" is not considered.

Each of the $10$ balls can choose which one of the boxes to go to. Since there are $4$ boxes, the ball has $4$ choices. This is true for each of the $10$ balls, so the product rule tells us there are $$\underbrace{4 \times \cdots \times 4}_{10 \text{ times}} = 4^{10}$$ ways to place all the balls into the boxes.

Note it is possible for none of the balls to choose some box, so it is possible that some boxes are empty (but it can't happen that all boxes are empty).

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