Question

In the expansion of ${(1+x)}^{43}$, if the coefficients of ${(2r+1)}^{th}$ and ${(r+2)}^{th}$ terms are equal, then what is the value of $r(r\ne 1)$?

• A. $5$
• B. $14$
• C. $21$
• D. $22$

Answer 1

The correct option is B $14$

Given $(1+x{)}^{43}$, coefficients of $(2r+1{)}^{th}$ and $(r+2{)}^{th}$ terms are equal

Using General formula $\to T_{r+1}{=}^n{C}_r{a}^{n-r}{b}^r....\left(1\right)$

$(2r+1{)}^{th}$ term, put $r=2r$ in $\left(1\right)$

$T_{2r+1}{=}^n{C}_{2r}.\left(1{)}^{43-2r}\right(X{)}^{2r}$

$(r+2{)}^{th}$ term, put $r=r+1$ in $\left(1\right)$

$T_{r+2}{=}^n{C}_{r+1}\left(1{)}^{43-(r+1)}\right(1{)}^{r+1}$

Coefficients are Equal, therefore

${\Rightarrow }^{43}{C}_{2r}{=}^{43}{C}_{r+1}$

If ${}^n{C}_y{=}^n{C}_z$, then $y=z$ or $y=n-z$

$2r=r+1\Rightarrow r=1$ but $r\ne 1$

$\therefore 2r=43-(r+1)\to 3=42\to r=14$.