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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2018 · Mathematics · Binomial Theorem

Let the coefficient of the middle term of the binomial expansion of...

Let the coefficient of the middle term of the binomial expansion of (1+x)2n(1+x)^{2 \mathrm{n}} be α\alpha and those of two middle terms of the binomial expansion of (1+x)2n−1(1+x)^{2 \mathrm{n}-1} be β\beta and γ\gamma. Which one of the following relations is correct?

Options

  1. Aα>β+γ\alpha > \beta+\gamma
  2. Bα<β+γ\alpha < \beta+\gamma
  3. Cα=β+γ\alpha=\beta+\gamma
  4. Dα=βγ\alpha=\beta \gamma

Correct answer

C. α=β+γ\alpha=\beta+\gamma

Step-by-step solution

α=2nCnβ=2n−1Cnγ=2n−1Cn−1β+γ=2n−1Cn+2n−1Cn−1=2nCn=α\begin{aligned} & \alpha={ }^{2 n} \mathrm{C}_n \\ & \beta={ }^{2 n-1} \mathrm{C}_n \\ & \gamma={ }^{2 n-1} \mathrm{C}_{n-1} \\ & \beta+\gamma={ }^{2 n-1} \mathrm{C}_n+{ }^{2 n-1} \mathrm{C}_{n-1}={ }^{2 n} \mathrm{C}_n=\alpha\end{aligned}
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