Expansion of $(1+x)^{2 n}$ is $1+^{2 n} C_{1} x+^{2 n} C_{2} x^{2}$

$+\ldots \ldots+^{2 n} C_{r} x^{r}+^{2 n} C_{r+1} x^{r+1}+\ldots \ldots+^{2 n} C_{2 n} x^{2 n}$

As given $^{2n}{{\text{C}}_{r + 2}}{ = ^{2n}}{{\text{C}}_{3r}}$

$ \Rightarrow \frac{{(2n)!}}{{(r + 2)!(2n – r – 2)!}} = \frac{{(2n)!}}{{(3r)!(2n – 3r)!}}$

$\Rightarrow(3 r) !(2 n-3 r) !=(r+2) !(2 n-r-2) !………(1)$

Now, put value of $n$ from the given choices.

Choice $(a)$ put $n=2 r+1$ in $(1)$

$LHS:$ $(3 r) !(4 r+2-3 r) !=(3 r) !(r+2) !$

$\mathrm{RHS}:(r+2) !(3 r) !$

$\Rightarrow \mathrm{LHS}=\mathrm{RHS}$