Let \( { }^{n} C_{r} \) denote the binomial coefficient of \( x^{r}...

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Let \( { }^{n} C_{r} \) denote the binomial coefficient of \( x^{r} \) in the expansion of \( (1+x)^{n} \).
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If \( \sum_{k=0}^{10}\left(2^{2}+2 k\right)^{n} C_{k}=\alpha \cdot 3^{10}+\beta \cdot 2^{10}, \alpha, \beta \in \mathrm{R} \), then \( \alpha+\beta \) is equal to
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