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Who discovered the following theorem

$$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$

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    $\begingroup$ I'm willing to bet it was 'discovered' more than once. $\endgroup$ – Carl Witthoft Jul 31 '18 at 11:53
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I firstly encountered this identity while working with related sums. Within this question on MSE, asked by myself, the user Robert Z refered to the particular case you are asking for as Tepper's Identity. In order to be precise Tepper's Identity is a more general result given by

$$n!~=~\sum_{k=0}^n(-1)^k\binom nk (a-k)^n,a\in\mathbb R$$

The aforementioned proof of Tepper's Identity was given by Ravi Prakash and was posted on the $2$nd of January $2018$ on the official website of the Romanian Mathematical Magazine which can be found here. Later on it reappeared within RMM - Famous Inequalities Marathon 1 - 100 as problem $97$. There are various proofs avaible nowadays, for instance Another Proof of Tepper's Identity by F. J. Papp published in the Mathematics Magazine Vol. $45$, No. $3$ $($May, $1972)$, pp. $119-121$ and A Proof of Tepper’s identity by Nguyen Trung Tuan posted on the $10$th of October $2007$.

Concerning the origin it seems like all sources are dating back to an arcticle called A Factorial Conjecture by Myron Tepper published in the Mathematics Magazine Vol. $38$, No. $5$ $($Nov., $1965)$, pp. $303-304$ $($the index to this volume can be found here$)$ proposing the identity, perhaps, for the first time.

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