Made Easy FLT

Question

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??

 

Comments

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 :)