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Q.
Suppose $\displaystyle\sum_{r=0}^{2023} r^{22023} C_r=2023 \times \alpha \times 2^{2022}$. Then the value of $\alpha$ is
JEE MainJEE Main 2023Permutations and Combinations
Solution:
using result
$\displaystyle\sum_{ r =0}^{ n } r ^2\,\,{ }^{ n } C _{ r }= n ( n +1) \cdot 2^{ n -2}$
Then $\displaystyle\sum_{ r =0}^{2023} r ^2\,\,{ }^{2023} C _{ r }=2023 \times 2024 \times 2^{2021}$
$ =2023 \times \alpha \times 2^{2022} \text { So, } $
$ \Rightarrow \alpha=1012$