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How many ways are there to put 4 balls in 3 boxes if the balls are distinguishable but the boxes are not?

 May 17, 2020
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If the ball are distinguishable, let's say that they are red (R), yellow (Y), green (G), and blue (B).

The boxes are not distinguishable.

 

We could put all four balls in one box; this can be done in only one way [R,Y,G,B].

Since the boxes are not distinguishable, it doesn't make any difference in what box we place

the four balls.

 

We could put three balls in one box, and one ball in another box. This can be done 4 ways:

[R,Y,G] - [B]     [R,Y,B] - [G]     [R,Y,G] - [B]     [ Y,B,G] - [R]

 

We can put two balls in one box, and two balls in another box. This can be done 3 ways:

[R,Y] - [G,B]     [R,G] - [B,Y]     [R,B] - [G,Y]

 

We can put two balls in one box, and one ball in each of the other two boxes. This can be

done 6 ways:

[R,Y] - [G] - [B]        [R,G] - [B] - [Y]          [R,B] - [G] - [Y]

[Y,B] - [R] - [G]         [Y,G] - [R] - [B]          [G,B] - [R] - [Y]

 May 17, 2020

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