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# Counting

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In how many ways can you distribute \$8\$ indistinguishable balls among \$5\$ distinguishable boxes, if at least three of the boxes must be empty?

May 30, 2024

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We can use casework to solve this problem.

If 3 boxes are empty, then there are two possibilites.

First, we could have 2 boxes with balls in them.

Case 1

Let's say one box and has 7 and the other has 1.

There are 2 ways we can do this.

Case 2

Now, let's say there is one boy with 6 and one box with 2,

There are also 2 ways we can do this.

Case 3

One box has 3 and the other has 5.

Another 2 ways we can do this.

Case 4

One box has 4 and the other has 4.

Only one way to do this.

\(2 * 2 * 2 * 1 = 8 \) ways we can do this in total

We choose 2 boxes out of the 5 we could have chosen from, which is just \(5 \choose 2\) = 10.

\(10 * 8 = 80\) ways to do 2 boxes.

Now, let's say only 1 box has all the balls. There are 5 ways to choose a box with all 8 balls in it.

\(80+5=85\). There are 85 ways we could do this!

Thanks! :)

May 30, 2024