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JEE Advanced 2023 Paper 2 · Mathematics · Binomial Theorem · medium

The value of S=Σk=0^8((-1)^kbinom8k)/(k+3) is

Single correct option

The value of S=∑k=08(−1)k(8k)k+3\displaystyle S=\sum_{k=0}^{8}\frac{(-1)^k\binom{8}{k}}{k+3} is
  1. A. 1495\dfrac1{495}
  2. B. 1165\dfrac1{165}
  3. C. 199\dfrac1{99}
  4. D. 1990\dfrac1{990}
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Answer: (A) 1495\dfrac1{495}

Use 1k+3=∫01xk+2dx\frac1{k+3}=\int_0^1x^{k+2}dx. Then S=∫01x2∑k=08(8k)(−x)k dx=∫01x2(1−x)8dxS=\int_0^1x^2\sum_{k=0}^8\binom8k(-x)^k\,dx=\int_0^1x^2(1-x)^8dx. This is a Beta integral: B(3,9)=Γ(3)Γ(9)Γ(12)=2⋅8!11!=211⋅10⋅9=1495B(3,9)=\frac{\Gamma(3)\Gamma(9)}{\Gamma(12)}=\frac{2\cdot8!}{11!}=\frac{2}{11\cdot10\cdot9}=\frac1{495}. Direct evaluation of the sum with exact fractions agrees. The other options come from slips in the factorials (for example, 1990\frac{1}{990} is half the correct value).
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