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If we do it with stirling approximation , then ans 301, but that is not correct according to ans key. But why cannot we solve it stirling approximation?? What is wrong in it??

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The Stirling numbers of the second kind $S(n,k)$ count the number of ways to partition a set of n labelled objects into k nonempty unlabelled subsets.

Here subsets(children) are labelled and objects(Balls) are unlabelled.
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thank u :)