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
Solution:
Instead of ${ }^{n} C_{k}$ it must be ${ }^{10} C_{k}$ i.e.
$\sum_{k=0}^{10}\left(2^{2}+3 k\right)^{10} C_{k}=\alpha .3^{10}+\beta .2^{10}$
$\mathrm{LHS}=4 \sum_{\mathrm{k}-0}^{10}{ }^{10} \mathrm{C}_{\mathrm{k}}+3 \sum_{\mathrm{k}=0}^{10} \mathrm{k} \cdot \frac{10}{\mathrm{k}} \cdot{ }^{9} \mathrm{C}_{\mathrm{k}-1}$
$=4.2^{10}+3.10 .2^{9}$
$=19.2^{10}=\alpha \cdot 3^{10}+\beta \cdot 2^{10}$
$\Rightarrow \alpha=0, \beta=19 \Rightarrow \alpha+\beta=19$