Let ${ }^{n} C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+x)^{n}$. If $

Question

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

Professor PC Thomas Classes · Answer

(19) : Let $n=10$\ $	herefore \sum_{k=0}^{10}(3 k+4)^{10} C_{k}=30 \sum_{k=1}^{10}{ }^{9} C_{k-1}+4 \sum_{k=0}^{10}{ }^{10} C_{k}$\ $=30 \cdot 2^{9}+4 \cdot 2^{10}=19 \cdot 2^{10}$\ $\Rightarrow \alpha=0$ and $\beta=19$. Then, $\alpha+\beta=19$.\