Question

If an the expansion of ${\left(1+x\right)}^{15}$, the coefficients of ${\left(2r+3\right)}^{\mathrm{th}}\mathrm{and}{\left(r-1\right)}^{\mathrm{th}}$ terms are equal, then the value of r is
(a) 5
(b) 6
(c) 4
(d) 3

Answer 1

(a) 5
$$\begin{gathered} \mathrm{Coefficients}\mathrm{of}(2r+3)\mathrm{th}\mathrm{and}(r-1)\mathrm{th}\mathrm{terms}\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{expansion}\mathrm{are}{}^{15}C_{2r+2}\mathrm{and}{}^{15}C_{r-2.} \\ \mathrm{Thus},\mathrm{we}\mathrm{have} \\ {}^{15}C_{2r+2}={}^{15}C_{r-2} \\ \Rightarrow 2r+2=r-2\mathrm{or}2r+2+r-2=15\left[\because \mathrm{if}{}^n{C}_x{=}^n{C}_y\Rightarrow x=y\mathrm{or}x+y=n\right] \\ \Rightarrow r=-4\mathrm{or}r=5 \\ \mathrm{Neglecting}\mathrm{the}\mathrm{negative}\mathrm{value},\mathrm{We}\mathrm{have} \\ r=5 \end{gathered}$$