If for positive integers $r 1,\ n 2$, the coefficient of the $(3r)$th and $(r + 2)$th powers of $x$ in the expansion of $(1 + x)^{2n}$ are equal, then

Question

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

Dr. Abubacker · Accepted Answer

(c)
In the expansion of $(1 + x)^{2n}$, the general term
$= \ ^{2n}C_{k}x^{k},\ 0 \leq k \leq 2n$
As given for $r > 1,\ n > 2,\ \ ^{2n}C_{3r} = \ ^{2n}C_{r + 2}$
$\Rightarrow$Either $3r = r + 2$ or $3r = 2n - (r + 2)\ \ \ \left( \because{\ ^{n}C}_{r} = {\ ^{n}C}_{n - r} \right)$
$\Rightarrow \ r = 1$ or $n = 2r + 1$
We take the relation only
$n = 2r + 1\ \ \ (\because r > 1)$

Q. If for positive integers $r > 1,\ n > 2$, the coefficient of the $(3r)$th and $(r + 2)$th powers of $x$ in the expansion of $(1 + x)^{2n}$ are equal, then

Explanation

Reviewed by Dr. Abubacker

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