Binomial Theorem Question 2
Question 2 - 24 January - Shift 1
Suppose $\sum _{r=0}^{2023} r^{2} \ { }^{2023} C_r=2023 \times \alpha \times 2^{2022}$. Then
the value of $\alpha$ is
Show Answer
Answer: (1012)
Solution:
Formula: Important Results on Binomial Coefficients (xvii)
using result
$\sum _{r=0}^{n} r^{2} \ { }^{n} C_r=n(n+1) \cdot 2^{n-2}$
Then $\sum_{\mathrm{r}=0}^{2023} \mathrm{r}^2 \ { }^{2023} \mathrm{C}_{\mathrm{r}}=2023 \times 2024 \times 2^{2021}$
$2023 \times 2024 \times 2^{2021}=2023 \times \alpha \times 2^{2022} \text { So, } \Rightarrow \alpha=1012 $
