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