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Condition between n-1Cr and nCr+1 with a parameter k

This type of question, asked 2 times: 27 Jan 2024, 2 Apr 2026 S1. All 2 questions below, oldest first.

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  1. Question 1

    27 Jan 2024

    61
     n−1Cr=(k2−8) nCr+1\ ^{n- 1}C_{r}=\left( k^{2}- 8 \right)\ ^{n}C_{r+ 1} if and only if:

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  2. Question 2

    2 Apr 2026 S1

    57
    The number of elements in the set S={(r,k):k∈ZS=\left\{ (r,k):k\in Z \right. and   36Cr+1=6( 35Cr)(k2−3)}\left. \ \ ^{36}C_{r+ 1}=\frac{6\left( \ ^{35}C_{r} \right)}{\left( k^{2}- 3 \right)} \right\}, is

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