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Hockey-stick sums of binomial coefficients

This type of question, asked 4 times: 2021 Paper 2, 26 Jun 2022, 25 Jan 2023, 4 Apr 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    77
    If n≥2n\geq 2 is a positive integer, then the sum of the series  n+1C2+2( 2C2+ 3C2+ 4C2+….+ nC2)\ ^{n + 1}C_{2}+ 2\left( \ ^{2}C_{2}+\ ^{3}C_{2}+\ ^{4}C_{2}+ \ldots. +\ ^{n}C_{2} \right) is :

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  2. Question 2

    26 Jun 2022

    175
    If ( 40C0)+( 41C1)+( 42C2)+…+( 60C20)=mn 60C20, m\left( \ ^{40}C_{0} \right)+\left( \ ^{41}C_{1} \right)+\left( \ ^{42}C_{2} \right)+ \ldots +\left( \ ^{60}C_{20} \right)=\frac{m}{n}\ ^{60}C_{20},\text{ }m and nn are coprime, then m+nm + n is equal to

  3. Question 3

    25 Jan 2023

    156
    ∑k=0645+kC3\sum_{k=0}^{6} {}^{45+k}C_{3} is equal to

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  4. Question 4

    4 Apr 2026 S1

    59
    Let the smallest value of k∈Nk \in\mathbb{N}, for which the coefficient of x3x^{3} in (1+x)3+(1+x)4+(1+x)5+…(1 + x)^{3}+ (1 + x)^{4}+ (1 + x)^{5}+ \ldots +(1+x)99+(1+kx)100,x≠0+ (1 +x)^{99}+ (1 +kx)^{100},x\neq 0, is (43n+1014)\left( 43n+\frac{101}{4} \right) (  100C3\ ^{100}C_{3} ) for some n∈Nn \in\mathbb{N}, be pp. Then the value of pp +n is:

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