+0  
 
0
2
1
avatar+388 

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
 #1
avatar+810 
+1

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

4 Online Users

avatar
avatar
avatar
avatar