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General Term of Binomial Expansion

If for positi...

Question

If for positive integers r > 1 n > 1 and the coefficient of $(3 r)^{t h}$ and $(r + 2)^{t h}$ terms in the binomial expansion of $(1 + x)^{2 n}$ are equal, then

n = 2r+1

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n=3r

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n =2r-1

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n=2r

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Solution

The correct option is D
n=2r

$T_{3 r} =^{2 n} C_{3 r - 1} (x)^{3 r - 1}$
$T_{r + 2} =^{2 n} C_{r + 1} (x)^{r + 1}$
It is given the coefficients are equal.
$\Rightarrow^{20} C_{r + 1} =^{20 C_{3 r - 1}}$
$\Rightarrow 3 r - 1 = r + 1 o r (3 r - 1) + (r + 1) = 2 n$
$\Rightarrow r = 1 o r n = 2 r$
But r > 1
$\Rightarrow n = 2 r$

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If for positive integers r > 1 n > 1 and the coefficient of (3r)th and (r+2)th terms in the binomial expansion of (1+x)2n are equal, then

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If for positive integers r > 1 n > 1 and the coefficient of $(3 r)^{t h}$ and $(r + 2)^{t h}$ terms in the binomial expansion of $(1 + x)^{2 n}$ are equal, then