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

$\begin{array}{1 1}(A)\;n=2r\\(B)\;n=3r\\(C)\;n=2r+1\\(D)\;\text{None of these}\end{array}$

Toolbox:
• $T_{r+1}=nC_r a^{n-r}b^r$
$T_{r+2}=2nC_{r+1}x^{r+1}$
$T_{3r}=2nC_{3r-1}x^{3r-1}$
$\therefore 2nC_{r+1}=2nC_{3r-1}$
$\Rightarrow r+1=3r-1$
$\Rightarrow (r+1)+(3r-1)=2n$
$\Rightarrow 4r=2n$ or $r=\large\frac{n}{2}$
$\Rightarrow r=\large\frac{n}{2}$
$\Rightarrow n=2r$
Hence (A) is the correct answer.