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Let $k \leq n$ be positive integers. I would like a proof of the following identity:

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

EDIT: For context, I am using this to verify the below polarization identity for writing the multilinear form $\overline{Q}$ associated to a degree-$k$ homogeneous polynomial $Q$ (Equation taken from Landsberg's book Tensors:Geometry and Applications). In particular, I need to verify this identity in order to show that the proposed multilinear form $\overline{Q}$ agrees with the polynomial $Q$, i.e. that $\overline{Q}(v,\dots, v)=Q(v)$ for all $v$ in the underlying vector space.

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I verified my desired identity in Mathematica, but I don't know how to prove it.

Ben
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  • Why was this downvoted? – Ben Apr 17 '22 at 20:32
  • Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – Shaun Apr 17 '22 at 20:52
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    @Shaun Done. Thanks – Ben Apr 17 '22 at 21:24
  • I checked and I get $n!$ in both cases... Can you check again? – Ben Apr 17 '22 at 21:49
  • To Ben: You're right, my apologies, I made clerical mistakes. I deleted my previous comment. – user2661923 Apr 17 '22 at 22:29

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