Question

If the coefficients of $\left(2r+4\right)\mathrm{th}\mathrm{and}\left(r-2\right)$th terms in the expansion of ${\left(1+x\right)}^{18}$ are equal, find r.

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ (1+x{)}^{18} \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{coefficient}\mathrm{of}\mathrm{the}r\mathrm{th}\mathrm{term}\mathrm{in}\mathrm{the}\mathrm{expansion}\mathrm{of}(1+x{)}^n\mathrm{is}{}^{n}C_{r-1} \\ \mathrm{Therefore},\mathrm{the}\mathrm{coefficients}\mathrm{of}\mathrm{the}(2r+4)\mathrm{th}\mathrm{and}(r-2)\mathrm{th}\mathrm{term}s\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{expansion}\mathrm{are}{}^{18}C_{2r+4-1}\mathrm{and}{}^{18}C_{r-2-1} \\ \mathrm{For}\mathrm{these}\mathrm{coefficients}\mathrm{to}\mathrm{be}\mathrm{equal},\mathrm{we}\mathrm{must}\mathrm{have} \\ {}^{18}C_{2r+3}={}^{18}C_{r-3} \\ \Rightarrow 2r+3=r-3\mathrm{or},2r+3+r-3=18[\because {}^{n}C_r={}^{n}C_s\Rightarrow r=s\mathrm{or}r+s=n] \\ \Rightarrow r=-6\mathrm{or},r=6 \\ \mathrm{Neglecting}\mathrm{negative}\mathrm{value}\mathrm{We}\mathrm{get} \\ r=6 \end{gathered}$$