I have stumbled across the formula $ \sum\limits_{k=1}^n \binom{n}{k} \cdot (n-k)! \cdot k!$ in some applied context. It looks as though there might be a nice combinatorial background to this formula. Is there another way to express it? Maybe a term that only depends on n? I am thinking of the famous equation expressing $ 2^n$ as a sum of binomial coefficients and I am hoping that this may be a similar (albeit more complicated) instance. TIA!
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