In how many ways can 4 balls be placed in 4 boxes if the balls are distinguishable, and the boxes are indistinguishable?

Jasonkiln Aug 27, 2022

#1**+1 **

This is calculated using "Stirling Numbers of the Second Kind"

**k=4; n=4; (1/k! * sumfor(i, 0, k,(-1)^i * (k nCr i) * (k - i)^n) * k!**

**==24 different ways - without any restrictions**

Guest Aug 27, 2022

edited by
Guest
Aug 27, 2022

#2**0 **

Hey I don't think 24 would be the correct answer

Also do you think there is another way to do it? I've never seen this before.

Guest Aug 27, 2022

#4**0 **

There are 5 cases:

4 - 0 - 0 - 0

3 - 1 - 0 - 0

2 - 2 - 0 - 0

2 - 1 - 1 - 0

1 - 1 - 1 - 1

For the first case, there is only 1 way.

In the second case, there are \({4 \choose 3} = 4 \) ways.

In the third case, there are \({4 \choose 2} = 6\) ways.

In the fourth case, there are \({4 \choose 2} \times 2 = 12\) ways.

For the fifth case, there is only 1 way.

So, there are \(1 + 4 + 6 + 12 + 1 = \color{brown}\boxed{24}\) ways.

BuilderBoi Aug 28, 2022