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Q.
If for positive integers $r>\, 1, n >\, 2,$ the coefficients of the $\left(3r\right)^{th}$ and $\left(r+2\right)^{th}$ powers of x in the expansion of $\left(1+x\right)^{2n}$ are equal, then $n$ is equal to:
Expansion of $\left(1+x\right)^{2n} is\, 1+ ^{2n}C_{1} x+^{2n}C_{2} x^{2}$
$+......+ ^{2n}C_{r}x^{r} +^{2n}C_{r+1} x^{r+1}+......+ ^{2n}C_{2n} x^{2n}$
As given $^{2n}C_{r+2} =^{2n}C_{3r}$
$\Rightarrow \frac{\left(2n\right)!}{\left(r+2\right)!\left(2n-r-2\right)!}=\frac{\left(2n\right)!}{\left(3r\right)!\left(2n-3r\right)!}$
$\Rightarrow \left(3r\right)! \left(2n-3r\right)!=\left(r+2\right)! \left(2n-r-2\right)! \ldots\left(1\right)\quad$
Now, put value of n from the given choices.
Choice $(a)$ put $n = 2r +1 in (1)$
LHS : $(3r)! (4r+2-3r)! = (3r)! (r+2)!$
$RHS : (r+2) ! (3r)! $
$\Rightarrow LHS=RHS\quad$