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Sum of r squared times nCr from r = 0 to n

The same question, asked 3 times: 2021 Paper 18*, 24 Jan 2023*, 4 Apr 2025. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 18

    69
    If  20Cr\ ^{20}C_{r} is the co-efficient of xrx^{r} in the expansion of (1+x)20(1 +x)^{20}, then the value of ∑r=020r2 20Cr\sum_{r= 0}^{20} r^{2}\ ^{20}C_{r} is equal to :

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  2. Question 2

    24 Jan 2023

    84
    Suppose ∑r=02023r2 2023Cr=2023×α×22022\sum_{r = 0}^{2023} r^{2}\ ^{2023}C_{r}= 2023 \times \alpha \times2^{2022}. Then the value of α\alpha is

  3. Question 3

    4 Apr 2025

    138
    If 12⋅( 15C1)+22⋅( 15C2)+32⋅( 15C3)1^{2}\cdot\left( \ ^{15}C_{1} \right)+2^{2}\cdot\left( \ ^{15}C_{2} \right)+3^{2}\cdot\left( \ ^{15}C_{3} \right) +….+152⋅( 15C15)=+ \ldots. +15^{2}\cdot\left( \ ^{15}C_{15} \right)= 2m⋅3n⋅5k2^{m}\cdot3^{n}\cdot5^{k}, where m,n,k∈Nm,n,k \in N, then m+n+km + n + k is equal to :-

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