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# Distinguishability and Indistinguishability

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1. In how many ways can \(4\) b***s be placed in \(3\) boxes if the b***s are indistinguishable, and the boxes are distinguishable?

2. In how many ways can \(4\) b***s be placed in \(3\) boxes if the b***s are distinguishable, and the boxes are indistinguishable?

These are the only two types of distinguishability and indistinguishability I can't do... please explain!

Thanks!

Nov 10, 2023

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Problem 1

If the 4 balls are indistinguishable and the boxes are distinguishable, then the only way to count the number of ways to place the balls in the boxes is to consider the number of balls in each box.

There are 3 ways to place 4 balls in 3 boxes such that there are 4 balls in one box and 0 balls in the other two boxes.

There are 3 ways to place 4 balls in 3 boxes such that there are 3 balls in one box and 1 ball in another box.

There are 1 way to place 4 balls in 3 boxes such that there are 2 balls in each box.

Therefore, there are a total of 7​ ways to place 4 indistinguishable balls in 3 distinguishable boxes.

Problem 2

If the balls are distinguishable, but the boxes are indistinguishable, then we can treat the balls as if they were numbered 1, 2, 3, and 4.

There are 4 ways to place the ball numbered 1 into one of the three boxes.

Once we have placed the ball numbered 1, there are 3 ways to place the ball numbered 2 into one of the two remaining boxes.

Once we have placed the balls numbered 1 and 2, there are 2 ways to place the ball numbered 3 into one of the remaining boxes.

Finally, there is only 1 way to place the ball numbered 4 into the remaining box.

Therefore, there are a total of 4⋅3⋅2⋅1=24​ ways to place 4 distinguishable balls into 3 indistinguishable boxes.

Nov 10, 2023