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# Double Trouble 2.0

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1. How many ways are there to put 6 balls in 3 boxes if the balls are not distinguishable and neither are the boxes?

2. How many ways are there to put 6 balls in 3 boxes if the balls are not distinguishable but the boxes are?

Jun 25, 2019

#1
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1. How many ways are there to put 6 balls in 3 boxes if the balls are not distinguishable and neither are the boxes?

Assuming that  we can have empty boxes.....this boils down to the number of ways that we partition  6  identical balls into 3 identical boxes

These are

6, 0, 0

5, 1, 0

4, 2, 0

4, 1, 1

3, 3, 0

3, 2, 1

2, 2, 2

So......7 ways   Jun 26, 2019
edited by CPhill  Jun 26, 2019
#2
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2. How many ways are there to put 6 balls in 3 boxes if the balls are not distinguishable but the boxes are?

Again....assuming that we can have empty boxes.....the number of ways of distributing k  identical balls into n  distinguishable boxes is given by :

C ( k +n - 1 , n - 1)   =  C (6 +3 - 1, 3 - 1 )  = C (8,2)  =  28 ways   Jun 26, 2019