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In how many ways can $4$ balls be placed in $5$ boxes if the balls and boxes are both distinguishable?

 Dec 15, 2023
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When the boxes are indistinguishable, we only need to consider the number of balls placed in each box, not the specific box they are in. This essentially turns the problem into a combination problem.

 

Here's how to solve it:

 

Identify the total number of balls and boxes: We have 4 balls and 5 boxes.

 

Consider arrangements as partitions: Imagine dividing the 4 balls into groups, where each group represents the number of balls placed in a single box.

 

Use stars and bars: We can represent the partitions using stars and bars. For example, if we place 2 balls in the first box, 1 in the second, and 1 in the third, we can represent this as "* ** ** *" (3 bars and 2 stars).

 

Number of partitions: The number of stars and bars is related to the number of balls and boxes. In this case, we have 4 stars (representing balls) and 4 bars (representing separators between boxes). The total number of arrangements is given by the combination formula:

 

Number of arrangements = ⁸C₄ = (4 + 4)! / (4! * 4!) = 5

 

Therefore, there are 5 ways to place 4 distinguishable balls in 5 indistinguishable boxes. These arrangements correspond to:

 

All 4 balls in one box

3 balls in one box, 1 in another

2 balls in two different boxes

1 ball in each of 4 boxes

 

The specific boxes where the balls are placed don't matter as long as the distribution of balls across boxes remains the same.

 Dec 17, 2023

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