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Is there a binomial coefficient identity with alternating sums as

$$...= \sum_{k=0}^{n} (-1)^{k+n} ... $$ presented? With ... I mean something connected to binomial coefficients.

I searched many books and webpages but I could not find anything.

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In the calculus of finite differences, if $$ f(x) = \sum_{k=0}^n c_k\binom{x}{k} $$ then $$ c_k = \sum_{j=0}^k (-1)^{k-j}\binom{k}{j}f(j). $$

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