# The coefficient of $r^{th}$ and $(r+1)^{th}$ terms in the expansion of $(1+x)^{20}$ are in the ratio 1 : 2 then $r=$

$\begin{array}{1 1}(A)\;6\\(B)\;7\\(C)\;8\\(D)\;9\end{array}$

Toolbox:
• $T_{r+1}=nC_ra^{n-r} b^r$
$T_{r+1}=20C_rx^r$
$\therefore$ Coefficient of $T_{r+1}=20C_r$
$\therefore \large\frac{20C_{r-1}}{20C_r}=\large\frac{1}{2}$
$\Rightarrow \large\frac{20!}{(r-1)!(21-r)!}\times \frac{r!(20-r)!}{20!}=\frac{1}{2}$
$\Rightarrow \large\frac{r}{21-r}=\frac{1}{2}$
$\Rightarrow r=7$
Hence (B) is the correct answer.