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

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35
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In how many ways can 4 balls be placed in 4 boxes if the balls are indistinguishable, and the boxes are distinguishable?

Feb 6, 2023

#1
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The balls are all the same so the only differenciating thing is how many balls are in each individual box.

The boxes can have

0,0,0,4       3 possibilities

0,0,1,3       6 possibilities

0,0,2,2       3 possibilities

0,1,1,2       3 possibilities

1,1,1,1       1 possibilities

So that are 3 + 6 + 3 + 3 + 1 = 16 unique outcomes

Feb 6, 2023
#2
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You got the right idea but got your permutations mixed up.

The boxes can have

0,0,0,4       3 possibilities............4! / 3! ==4

0,0,1,3       6 possibilities............4! / 2! ==12

0,0,2,2       3 possibilities............4! / 2!.2! ==6

0,1,1,2       3 possibilities.............4! / 2! ==12

1,1,1,1       1 possibilities..............4! / 4! ==1

Total ==4  +  12  +  6  +  12  +  1 ==35 ways - which is the same as:

[4 + 4 - 1] C 4 ==7 C 4 ==35 ways

Guest Feb 6, 2023