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Coefficient of x^r in a sum like (a+x)^n + x(a+x)^(n-1) + ... + x^n

This type of question, asked 5 times: 25 Jun 2022, 30 Jan 2023, 31 Jan 2024, 9 Apr 2024, 28 Jan 2026 S2. All 5 questions below, oldest first.

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  1. Question 1

    25 Jun 2022

    157
    The coefficient of x101x^{101} in the expression (5+x)500+x(5+x)499+x2(5+x)498+…..x500(5 + x)^{500}+ x(5 + x)^{499}+x^{2}(5 + x)^{498}+ \ldots..x^{500}, x>0x > 0, is

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  2. Question 2

    30 Jan 2023

    77
    The coefficient of x301x^{301} in (1+x)500+x(1+x)499+x2(1+x)498+……..x500(1 + x)^{500}+ x(1 + x)^{499}+x^{2}(1 + x)^{498}+ \ldots\ldots..x^{500} is :

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  3. Question 3

    31 Jan 2024

    174
    Let the coefficient of xrx^{r} in the expansion of
     (x+3)n−1+(x+3)n−2(x+2)+ (x+3)n−3(x+2)2+……..+(x+2)n−1\begin{matrix} & \ (x + 3)^{n - 1}+ (x + 3)^{n - 2}(x + 2) + \\ & \ (x + 3)^{n - 3}(x + 2)^{2}+ \ldots\ldots.. + (x + 2)^{n - 1} \end{matrix}
    be αr\alpha_{r}. If ∑r=0nαr=βn−γn,β,γ∈N\sum_{r = 0}^{n} \alpha_{r}=\beta^{n}-\gamma^{n},\beta,\gamma \in N, then the value of β2+γ2\beta^{2}+\gamma^{2} equals

  4. Question 4

    9 Apr 2024

    66
    The coefficient of x70x^{70} in x2(1+x)98+x3(1+x)97+x^{2}(1 + x)^{98}+x^{3}(1 + x)^{97}+ x4(1+x)96+x^{4}(1 + x)^{96}+ +x54(1+x)46+x^{54}(1 + x)^{46} is  99Cp− 46Cq\ ^{99}C_{p}-\ ^{46}C_{q}. Then a possible value to p+qp + q is:

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  5. Question 5

    28 Jan 2026 S2

    52
    The sum of the coefficients of x499x^{499} and x500x^{500} in (1+x)1000+x(1+x)999+x2(1+x)998+……..+x1000(1 +x)^{1000}+x(1 +x)^{999}+x^{2}(1 +x)^{998}+ \ldots\ldots.. +x^{1000} is

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