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# Counting

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Help I'm dying

Find the number of ways of placing 4 balls in 5 boxes if the balls are indistinguishable and the boxes are distinguishable.

Jan 27, 2023

#1
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First lay out the number of ways to distribute the 4 balls:

4 - 0 - 0 - 0 - 0

3 - 1 - 0 - 0 - 0

2 - 1 - 1 - 0 - 0

2 - 2 - 0 - 0 - 0

1 - 1 - 1 - 1 - 0

For the first case, there are $${5 \choose 1} = 5$$ ways to choose the box for the 4, and the rest must be 0

For the second case, there are $${5 \choose 3} \times {2 \choose 1} = 20$$ ways.

For the third case, there are $${5 \choose 1} \times {4 \choose 2} = 30$$ cases

For the fourth case, there are $${5 \choose 2} = 10$$ cases.

For, the final one, there are $${5 \choose 1} = 5$$ ways.

So, in total there are $$5 + 5 + 10 + 20 + 30 = \color{brown}\boxed{70}$$ ways.

Jan 27, 2023
#2
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