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How many ways are there to put 5 balls in 3 boxes if the balls are not distinguishable but the boxes are?

 Feb 2, 2023
 #1
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There are 10 ways to put 5 balls in 3 boxes. You can either have all the balls in one box, all but one ball in one box, or one ball in each box. Here are the combinations:

5 balls in one box, 0 in the other two

4 balls in one box, 1 in another box, 0 in the last box

4 balls in one box, 0 in another box, 1 in the last box

3 balls in one box, 2 in another box, 0 in the last box

3 balls in one box, 1 in another box, 1 in the last box

3 balls in one box, 0 in another box, 2 in the last box

2 balls in one box, 2 in another box, 1 in the last box

2 balls in one box, 1 in another box, 2 in the last box

1 ball in one box, 2 in another box, 2 in the last box

0 balls in one box, 3 in another box, 2 in the last box

 

This gives us a total of 5 + 4 + 4 + 3 + 3 + 3 + 2 + 2 + 1 = 27 ways.

 Feb 2, 2023
 #2
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[5 + 3 - 1] C [5] = 21 ways

 

Source: https://en.wikipedia.org/wiki/Twelvefold_way

              [Bottom table, item # 7]

 Feb 2, 2023

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