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In how many ways can you distribute 10 indistinguishable balls among 8 distinguishable boxes, if at least one of the boxes must be empty?

 Dec 26, 2022
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[10 + 8 -1] C [8 - 1] ==17 C 7==19,448 - total number of ways.

 

Almost all of the above 19,448 ways have at least 1 empty box with the exception of:

 

8 C 2 ==28 boxes that have no empty boxes as follows:

 

11111122 , 11111212 , 11111221 , 11112112 , 11112121 , 11112211 , 11121112 , 11121121 , 11121211 , 11122111 , 11211112 , 11211121 , 11211211 , 11212111 , 11221111 , 12111112 , 12111121 , 12111211 , 12112111 , 12121111 , 12211111 , 21111112 , 21111121 , 21111211 , 21112111 , 21121111 , 21211111 , 22111111 , Total =  28 ways with no empty boxes.

 Dec 27, 2022

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