Binomial Chapter :: Help

How to solve this:

Find The value of :: $\displaystyle{\sum _{ 0 }^{ n }{ { { ^{ n }{ P } } }_{ r } }}$ .

here P(n,r) means permutation of n things taken r at a time .

I have not study Binaomial chapter in detaied Yet . But I need this sumition whlile solving an question of PNC .

PLease Help , I have no idea for such summitions , Please Explain , Thanks a lot

#Help

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Comments

We can actually evaluate the sum as $\Gamma(n+1,1)$ , where $\Gamma(a,x)$ is the Upper Incomplete Gamma function .

$\begin{aligned} \sum_{r=0}^{n} \binom{n}{r} r! &= \sum_{r=0}^{n} \binom{n}{r} \int_{0}^{\infty} e^{-x} x^r \mathrm{d}x\\ &= \int_{0}^{\infty} \sum_{r=0}^{n} \binom{n}{r} e^{-x} x^r \mathrm{d}x\\ &= \int_{0}^{\infty} e^{-x} (1+x)^{n} \mathrm{d}x\\ &=\boxed{e \cdot \Gamma(n+1,1)}\end{aligned}$ .

This solution has been inspired from the genius Pratik Shastri's solution to this problem .