Closed form of $n!\sum_{k=3}^{n-1}{{n-2}\choose{k-1}}$

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$n$ is given, and it takes part in the following formula. $$n!\sum_{k=3}^{n-1}{{n-2}\choose{k-1}}$$

Is there a nicer way for expressing it? Without the summation sign?

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Hint: Consider the binomial expansion of $(1+1)^{n-2}$.