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Let $n$ and $k$ be integers with $1 \leq k \leq n.$ Show that $\displaystyle{}\sum_{k=1}^{n} \binom{n}{k}\binom{n}{k − 1} = \dfrac{\binom{2n + 2}{n + 1}}{2} − \binom{2n}{n}.$
in Combinatory

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