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# In how many ways can 4 balls be placed in 4 boxes if the balls are distinguishable, and the boxes are indistinguishable?

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In how many ways can 4 balls be placed in 4 boxes if the balls are distinguishable, and the boxes are indistinguishable?

Aug 27, 2022

#1
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This is calculated using "Stirling Numbers of the Second Kind"

k=4; n=4; (1/k! * sumfor(i, 0, k,(-1)^i * (k nCr  i) * (k - i)^n) * k!

==24 different ways - without any restrictions

Aug 27, 2022
edited by Guest  Aug 27, 2022
#2
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Hey I don't think 24 would be the correct answer
Also do you think there is another way to do it? I've never seen this before.

Aug 27, 2022
#3
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Go to this link and see the formula used (Formula(9)) under "distinct"   and   "identical" :

https://www.careerbless.com/aptitude/qa/permutations_combinations_imp7.php

Guest Aug 27, 2022
#4
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There are 5 cases:

4 - 0 - 0 - 0

3 - 1 - 0 - 0

2 - 2 - 0 - 0

2 - 1 - 1 - 0

1 - 1 - 1 - 1

For the first case, there is only 1 way.

In the second case, there are $${4 \choose 3} = 4$$ ways.

In the third case, there are $${4 \choose 2} = 6$$ ways.

In the fourth case, there are $${4 \choose 2} \times 2 = 12$$ ways.

For the fifth case, there is only 1 way.

So, there are $$1 + 4 + 6 + 12 + 1 = \color{brown}\boxed{24}$$ ways.

Aug 28, 2022