I'm struggling to understand how to approach this question to begin with. I think there are a total of 150 toys (since there are 10 of each of the 15 kinds of toys) but otherwise I don't know where to start from here. Help would be greatly appreciated!
2026-04-08 22:21:29.1775686889
Ways to distribute 15 distinct kinds of toys with at least 10 of each kind to 10 kids so that each kid gets at most one kind of each toy?
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I am going to assume that the children are distinguishable - so giving toy A to Alice and toy B to Bob is a different distribution from giving toy B to Alice and toy A to Bob. However, I am going to assume that two toys of the same kind are not distinguishable.
I am also going to assume that there is no lower limit on the number of toys each child has i.e. some children may have no toys.
To have no more than one of each kind of toy, each child must have some subset of the 15 kinds of toys. There are $2^{15}$ different subsets if having no toys is a valid outcome. If every child has a toy of every kind then 10 toys of each kind are required - but we know there are at least 10 toys of each kind, so this is a valid distribution.
Assuming the children are distinguishable and there is no lower limit on the number of toys each child has, then the total number of distributions is $(2^{15})^{10} = 2^{150}$.