JEE Main 2021 (Online) 18th March Evening ShiftQuestion 1
Let ${}^{n}C_r$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+x)^{n}$. If $\sum_{k=0}^{10}\left(2^{2}+3k\right){}^{10}C_k=\alpha\cdot3^{10}+\beta\cdot2^{10}$, $\alpha,\beta\in\mathbf{R}$, then $\alpha+\beta$ is equal to $\_\_\_\_\_\_$ .
Binomial Theorem · Properties and Sums of Binomial Coefficients
Answer$19$
Solution
Concept. Split $(4+3k)\,{}^{10}C_k$ and use the two standard sums. Formulas used. (1) $\sum{}^{n}C_k=2^n$ and $\sum k\,{}^{n}C_k=n\,2^{n-1}$Step 1 - Split. By (1) with $n=10$: $$\sum_{k=0}^{10}(4+3k)\,{}^{10}C_k=4\cdot2^{10}+3\cdot10\cdot2^9=4\cdot2^{10}+15\cdot2^{10}=19\cdot2^{10}.$$ Step 2 - Compare. No power of $3$ appears, so $\alpha=0$ and $\beta=19$. Step 3 - Value. $\alpha+\beta=19$. Answer: 19