Suppose ∑ r = 0 2023 r 2 · C r 2023 = 2023 × α × 2 2022 , then the value of α…

Given,

$\sum _{r=0}^{2023}{r}^2·{}^{2023}C_r=2023\times \alpha \times 2^{2022}$

Now taking L.H.S we get,

$\sum _{r=0}^{2023}{r}^2·{}^{2023}C_r=\sum _{r=1}^{2023}{r}^2·\frac{2023}{r}·{}^{2022}C_{r-1}$

$\Rightarrow \sum _{r=0}^{2023}{r}^2·{}^{2023}C_r=\sum _{r=1}^{2023}r·2023·{}^{2022}C_{r-1}$

$\Rightarrow \sum _{r=0}^{2023}{r}^2·{}^{2023}C_r=2023\sum _{r=1}^{2023}\left(r-1+1\right)·{}^{2022}C_{r-1}$

$\Rightarrow \sum _{r=0}^{2023}{r}^2·{}^{2023}C_r=2023\left[\sum _{r=1}^{2023}\left(r-1\right)·{}^{2022}C_{r-1}+\sum _{r=1}^{2023}{}^{2022}C_{r-1}\right]$

$\Rightarrow \sum _{r=0}^{2023}{r}^2·{}^{2023}C_r=2023\left[2022\sum _{r=2}^{2023}{}^{2021}C_{r-2}+\sum _{r=1}^{2023}{}^{2022}C_{r-1}\right]$

$\Rightarrow \sum _{r=0}^{2023}{r}^2·{}^{2023}C_r=2023\left[2022\times 2^{2021}+2^{2022}\right]$

$\Rightarrow \sum _{r=0}^{2023}{r}^2·{}^{2023}C_r=2023\times 2^{2022}\left[1011+1\right]$

$\Rightarrow \sum _{r=0}^{2023}{r}^2·{}^{2023}C_r=2023\times 1012\times 2^{2022}$

Now on comparing with $\sum _{r=0}^{2023}{r}^2·{}^{2023}C_r=2023\times \alpha \times 2^{2022}$ we get,

$\alpha =1012$