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

Correct answer: (C) α=β+γ. α=^2 n C_n β=^2 n-1 C_n γ=^2 n-1 C_n-1 β+γ=^2 n-1 C_n+^2 n-1 C_n-1=^2 n C_n=α

Q174 / 270 · NDA 2018 (Phase 2) · Mathematics

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

A
$\alpha \gt \beta+\gamma$
B
$\alpha \lt \beta+\gamma$
Correct answer
C
$\alpha=\beta+\gamma$
D
$\alpha=\beta \gamma$

Solution

$\begin{aligned} & \alpha={ }^{2 n} \mathrm{C}_n \\ & \beta={ }^{2 n-1} \mathrm{C}_n \\ & \gamma={ }^{2 n-1} \mathrm{C}_{n-1} \\ & \beta+\gamma={ }^{2 n-1} \mathrm{C}_n+{ }^{2 n-1} \mathrm{C}_{n-1}={ }^{2 n} \mathrm{C}_n=\alpha\end{aligned}$

Hence, the correct answer is option: 'C'.

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

Solution

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