### 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 $\sum _{r=0}^{2023} r^2 \ { }^{2023} C _r=2023 \times 2024 \times 2^{2021}$

$2023 \times 2024 \times 2^{2021}=2023 \times \alpha \times 2^{2022} \text { So, } \Rightarrow \alpha=1012 $