Question

In the expansion of $(1+x{)}^{18}$ , if the coefficients of $(2r+4{)}^{th}$ and $(r-2{)}^{th}$ terms are equal, then the value of r is :

• A. 12
• B. 10
• C. 8
• D. 6

Answer 1

Answer: (D)

Since the coefficient of $(r+1{)}^{th}$ term in the expansion of ${\left(1+x\right)}^n{=}^n{C}_r$
$\therefore$ In the expansion of ${\left(1+x\right)}^{18}$
coefficient of ${\left(2r+4\right)}^{th}$ term ${=}^{18}{C}_{2r+3}$,
Similarly, coefficient of ${\left(r-2\right)}^{th}$ term in the expansion of ${\left(1+x\right)}^{18}{=}^{18}{C}_{r-3}$
If ${}^n{C}_r{=}^n{C}_s$ then $r+s=n$
So, ${}^{18}{C}_{2r+3}{=}^{18}{C}_{r-3}$ gives
$2r+3+r-3=18$
$\Rightarrow 3r=18\Rightarrow r=6.$