Question

If the coefficient of $(r+4)$th term and $(2r+1)$th term in the expansion of ${\left(1+x\right)}^{18}$ are equal, then $r$=

• A. $3$
• B. $5$
• C. $3$ or $5$
• D. None of these

Answer 1

Answer: (C)

$3$ or $5$

$(1+x{)}^{18}{=}^{18}{C}_6{x}^0{+}^{18}{C}_{1}x+----{-}^{18}{C}_n{x}^n+----$

coefficient of $r^{th}$ term ${=}^{18}{C}_{r-1}$

coefficient of $(r+4{)}^{th}$ term ${=}^{18}{C}_{r+4-1}{=}^{18}{C}_{r+3}----\left(1\right)$

coefficient of $(2r+1{)}^{th}$ term ${=}^{18}{C}_{2r+1-1}{=}^{18}{C}_{2r}----\left(2\right)$

Given:- $\left(1\right)=\left(2\right)$

${}^{18}{C}_{r+3}{=}^{18}{C}_{2r}$

$\frac{18!}{(18-(r+3\left)\right)!(r+3)!}=\frac{18!}{(18-2r)!\left(2r\right)!}$

$\Rightarrow r+3+2r=18$

$\Rightarrow 3r+3=18$

$\Rightarrow r=5$

Or

$r+3=2r$

$\Rightarrow r=3$

$r=3$

$\therefore r=5,3$