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Sums of nCr/(r+1) and similar, using integration

This type of question, asked 6 times: 12 Apr 2023, 29 Jan 2024, 22 Jan 2025, 22 Jan 2026 S2, 23 Jan 2026 S1, 8 Apr 2026 S2. All 6 questions below, oldest first.

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  1. Question 1

    12 Apr 2023

    76
    If 1n+1 nCn+1n nCn−1\frac{1}{n + 1}\ ^{n}C_{n}+\frac{1}{n}\ ^{n}C_{n - 1} +…+12 nC1+ nC0=102310+ \ldots +\frac{1}{2}\ ^{n}C_{1}+\ ^{n}C_{0}=\frac{1023}{10} then nn is equal to

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  2. Question 2

    29 Jan 2024

    89
    If  11C12+ 11C23+…..+ 11C910=nm\frac{\ ^{11}C_{1}}{2}+\frac{\ ^{11}C_{2}}{3}+ \ldots.. +\frac{\ ^{11}C_{9}}{10}=\frac{n}{m} with gcd(n,m)=1gcd(n,m) = 1, then n+mn + m is equal to

  3. Question 3

    22 Jan 2025

    72
    If ∑r=05 11C2r+12r+2=mn,gcd(m,n)=1\sum_{r = 0}^{5} \frac{\ ^{11}C_{2r + 1}}{2r + 2}=\frac{m}{n},gcd(m,n) = 1, then m−nm - n is equal to

  4. Question 4

    22 Jan 2026 S2

    70
    Let CrC_{r} denote the coefficient of xrx^{r} in the binomial expansion of (1+x)n,n∈N,0≤r≤n(1 + x)^{n},n \in N,0 \leq r \leq n. If Pn=C0−C1+223C2−234C3+…..+(−2)nn+1CnP_{n}=C_{0}-C_{1}+\frac{2^{2}}{3}C_{2}-\frac{2^{3}}{4}C_{3}+ \ldots.. +\frac{( - 2)^{n}}{n + 1}C_{n}, then the value of ∑n=1251P2n\sum_{n = 1}^{25} \frac{1}{P_{2n}} equals.

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  5. Question 5

    23 Jan 2026 S1

    62
    The value of  100C5051+ 100C5152+….+ 100C100101\frac{\ ^{100}C_{50}}{51}+\frac{\ ^{100}C_{51}}{52}+ \ldots. +\frac{\ ^{100}C_{100}}{101} is :

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  6. Question 6

    8 Apr 2026 S2

    58
    If 26(233( 13C2)+255( 12C4)+277( 12C6)+...+21313( 12C12))=313−α26\left( \frac{2^{3}}{3}\left( \ ^{13}C_{2} \right)+\frac{2^{5}}{5}\left( \ ^{12}C_{4} \right)+\frac{2^{7}}{7}\left( \ ^{12}C_{6} \right)+ ... +\frac{2^{13}}{13}\left( \ ^{12}C_{12} \right) \right)=3^{13}-\alpha, then α\alpha is equal to :

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