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

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

ManipalManipal 2012

Solution:

$3 r$ th term in the expansion of $(1+x)^{2 n}$
$={ }^{2 n} C_{3 r-1} x^{3 r-1}$
and $(r+2)$ th term in the expansion of $(1+x)^{2 n}$
$={ }^{2 n} C_{r+1} x^{r+1}$
Given that the binomial coefficients of $(3 r)$ th
and $(r+2)$ th terms are equal.
$\therefore { }^{2 n} C_{3 r-1}={ }^{2 n} C_{r+1}$
$\Rightarrow 3 r-1=r+1$
or $2 n=(3 r-1)+(r+1)$
$\Rightarrow 2 r=2$ or $2 n=4 r$
$\Rightarrow r=1$ or $n=2 r$
But $r>1$
$\therefore n=2 r$