Question

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

Answer 1

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