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Sums of products of binomial coefficients (Vandermonde identity)

This type of question, asked 7 times: 2021 Paper 2, 2021 Paper 8, 2021 Paper 19, 2021 Paper 20, 26 Jul 2022, 24 Jan 2023, 30 Jan 2024. All 8 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    88
    For integers nn and rr, let (nr)={ nCr, if n≥r≥00, otherwise \begin{pmatrix} n \\ r \end{pmatrix}=\left\{ \begin{matrix} \ ^{n}C_{r}, & \text{~if~}n\geq r\geq 0 \\ 0, & \text{~otherwise~} \end{matrix} \right. The maximum value of kk for which the sum ∑i=0k(10i)(15k−i)+∑i=0k+1(12i)(13k+1−i)\sum_{i= 0}^{k} \begin{pmatrix} 10 \\ i \end{pmatrix}\begin{pmatrix} 15 \\ k-i \end{pmatrix}+\sum_{i= 0}^{k+ 1} \begin{pmatrix} 12 \\ i \end{pmatrix}\begin{pmatrix} 13 \\ k+ 1 -i \end{pmatrix} exists, is equal to

  2. Question 2

    2021 Paper 8

    64
    The value of ∑r=06( 6Cr⋅ 6C6−r)\sum_{r = 0}^{6} \left( \ ^{6}C_{r}\cdot\ ^{6}C_{6 - r} \right) is equal to:

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  3. Question 3

    2021 Paper 19

    86
    Let (nk)\begin{pmatrix} n \\ k \end{pmatrix} denote  nCk\ ^{n}C_{k} and [nk]={(nk),if 0≤k≤n0,otherwise\begin{bmatrix} n \\ k \end{bmatrix}=\left\{ \begin{matrix} \begin{pmatrix} n \\ k \end{pmatrix}, & \text{if } 0 \leq k \leq n \\ 0, & \text{otherwise} \end{matrix} \right. If Ak=∑i=09(9i)[1212−k+i]+∑i=08(8i)[1313−k+i]A_{k}=\sum_{i = 0}^{9} \begin{pmatrix} 9 \\ i \end{pmatrix}\begin{bmatrix} 12 \\ 12 - k + i \end{bmatrix}+\sum_{i = 0}^{8} \begin{pmatrix} 8 \\ i \end{pmatrix}\begin{bmatrix} 13 \\ 13 - k + i \end{bmatrix} and A4−A3=190pA_{4}-A_{3}= 190p, then pp is equal to:

  4. Question 4

    2021 Paper 20

    73
    ∑k=020( 20Ck)2\sum_{k = 0}^{20} \left( \ ^{20}C_{k} \right)^{2} is equal to:

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  5. Question 5

    26 Jul 2022

    154
     ∑i,j=0i≠jn nCi nCj\ \sum_{\begin{matrix} i,j = 0 \\ i \neq j \end{matrix}}^{n} \ ^{n}C_{i}\ ^{n}C_{j} is equal to

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  6. Question 6

    24 Jan 2023

    161
    If ( 30C1)2+2( 30C2)2+3( 30C3)2+….+30( 30C30)2=α60!(30!)2\left( \ ^{30}C_{1} \right)^{2}+ 2\left( \ ^{30}C_{2} \right)^{2}+ 3\left( \ ^{30}C_{3} \right)^{2}+ \ldots. + 30\left( \ ^{30}C_{30} \right)^{2}=\frac{\alpha 60!}{(30!)^{2}} then α\alpha is equal to :

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  7. Question 7

    24 Jan 2023

    75
    The value of ∑r=022 22Cr 23Cr\sum_{r = 0}^{22} \ ^{22}C_{r}\ ^{23}C_{r} is

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  8. Question 8

    30 Jan 2024

    174
    Let α=∑k=0n(( nCk)2k+1)\alpha=\sum_{k = 0}^{n} \left( \frac{\left( \ ^{n}C_{k} \right)^{2}}{k + 1} \right) and β=∑k=0n−1( nCk nCk+1k+2)\beta=\sum_{k = 0}^{n - 1} \left( \frac{\ ^{n}C_{k}\ ^{n}C_{k + 1}}{k + 2} \right). If 5α=6β5\alpha= 6\beta, then nn equals