Mathematics - Binomial Theorem Question with Solution | TestHub

Expansion of $(1+x{)}^{2n}$ is $1+{}^{2n}{\mathrm{C}}_{1}x+{}^{2n}{\mathrm{C}}_2{x}^2$ $+\dots \dots .+{}^{2n}{\mathrm{C}}_r{x}^n+{}^{2n}{\mathrm{C}}_{r+1}{x}^{n+1}+\dots \dots +{}^{2n}{\mathrm{C}}_{2n}{x}^{2n}$ As given ${}^{2n}{\mathrm{C}}_{r+2}={}^{2n}{\mathrm{C}}_{3r}$ $$\begin{array}{rl}& \Rightarrow \frac{(2n)!}{(r+2)!(2n-r-2)!}=\frac{(2n)!}{(3r)!(2n-3r)!}\\ & \Rightarrow (3r)!(2n-3r)!=(r+2)!(2n-r-2)!\end{array}$$ 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 : $(r+2)!(3r)!$ $$\Rightarrow \text{ LHS }=\text{ RHS }$$