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Find the number of ways of distributing 4 different balls among 4 identical boxes.

So I was thinking that it would be \(4^4/4\) since you have 4 choices for each ball but there are 4 we since the boxes are identical we're overcounting by a factor of 4. Is this right?

 Feb 8, 2020
 #1
avatar+107414 
+2

Find the number of ways of distributing 4 different balls among 4 identical boxes.

Counting questions are always tricky, but here is my thinking.

 

distribution of balls  
1,1,1,11 way 
2,1,1,04C2=6 ways 
2,2,0,04C2/2=3 ways 
3,1,0,04C3=4 ways 
4,0,0,01 way 
 Total = 15 ways 

 

So no, I do not think your way makes any sense.

Feel free to ask questions.

 

The red is my edit. Impasta found a fault.  Good logic Impasta!

 
 Feb 8, 2020
edited by Melody  Feb 8, 2020
 #5
avatar+33 
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So are you implying for 2-1-1-1, for example, that out of the four balls, we choose 2 out of four balls to be put in one box, and we don't need to account for permutations?

 
Impasta  Feb 8, 2020
 #6
avatar+33 
+1

Wait, I'm pretty sure doing it that way overcounts some things. For instance, doing \(\binom{4}{2}\) for 2-2-0-0 will count each way twice since you choosing two balls or you choosing the other two will result in the same balls in boxes. But building off of your idea, I think I can get the answer. Thanks!

 
Impasta  Feb 8, 2020
edited by Impasta  Feb 8, 2020
 #7
avatar+107414 
+1

Yes you are right. Excellent spotting!

I have edited my answer acordingly.

 
Melody  Feb 8, 2020
edited by Melody  Feb 8, 2020

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