Question

Find the value of $r$ if the coefficients of $(2r+4{)}^{th}$ term and $(r-2{)}^{th}$ term in the expansion of $(1+x{)}^{18}$ are equal.

• A. $8$
• B. $6$
• C. $4$
• D. $9$

Answer 1

Answer: (B)

$T_{2r+4}={}^{18}{C}_{2r+3}{\left(x\right)}^{2r+3}$ and
$T_{r-2}={}^{18}{C}_{r-3}{x}^{r-3}$
We are given that ${}^{18}{C}_{2r+3}={}^{18}{C}_{r-3}$
$\Rightarrow$ either $2r+3=r-3$ or
$2r+3=18-\left(r-3\right)$
$\Rightarrow$ either $r=-6$ or $3r=18$
$\Rightarrow r=6$ ($\because r$ cannot be $-ve$)