Level
3

Evaluate

\[\sum_{a=0}^\infty \sum_{b=0}^\infty \sum_{c=0}^a \sum_{d=0}^b \sum_{e=0}^d {n+a \choose a} {b-e \choose d-e} {a \choose c} {b \choose e} (-1)^{b+c} n^e c^{d-e}\]

Now before you enter your final answer -

If your final answer is a rational number and is in the form \(\frac{p}{q}\) where \(p\) and \(q\) are coprime to each other then your final answer should be \(\frac{p\pi}{q}\).

Otherwise If your final answer is an irrational number then your final answer should be \(99\)

It is an entry in the Troll King Contest.

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