# Starling Sum

Let $$A(n,k)=\sum_{S\in[n]^{k}}\frac{1}{\Pi(S)}$$.

Then $$A(200,197)$$ can be expressed as $$x\times10^{-363}$$. Find $$x$$ up to 3 decimal places.

Details and Notations

i. $$[n]$$ means the set $$\{1,2,...,n\}.$$

ii. $$[n]^{k}$$ means the set of all subsets of $$[n]$$ that have $$k$$ elements.

iii. $$\in$$ means belongs to, i.e, $$S\in[n]^{k}$$ means $$S$$ is a subset of $$[n]$$ with $$k$$ elements.

iv. $$\Pi(S)$$ denotes the product of the elements of $$S$$.

v. The sum is over all $$k$$-element subsets $$S$$ of $$[n]$$.

vi. By convection $$\Pi(\phi)=1$$ where $$\phi$$ denotes the empty set.

vii. One must know about Stirling Numbers in order to solve it.

Example

$$A(n,1)=\sum_{S∈[n]^{1}}\frac{1}{\Pi(S)}=1+\frac{1}{2}+\cdots+\frac{1}{n}$$

$$A(n,2)=\sum_{S∈[n]^{2}}\frac{1}{\Pi(S)}=1+\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{2\times4}+\cdots$$

Try other interesting combinatorics in my set Hard

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