Question

If the coefficients of the $r$th term and the $(r+1)$th term in the expansion of $(1+x)^{20}$ are in the ratio $1:2$, then $r$ equals

• A. 6
• B. 7
• C. 8
• D. 9

Hint:

$\dfrac{\binom{n}{r-1}}{\binom{n}{r}} = \dfrac{r}{n-r+1}$. Use this ratio formula directly to avoid computing individual binomial coefficients.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Coefficient of $r$th term $= \binom{20}{r-1}$; coefficient of $(r+1)$th term $= \binom{20}{r}$.
Step 2: Detailed Explanation:
$\dfrac{\binom{20}{r-1}}{\binom{20}{r}} = \dfrac{1}{2} \Rightarrow \dfrac{r}{21-r} = \dfrac{1}{2} \Rightarrow 2r = 21-r \Rightarrow 3r = 21 \Rightarrow r = 7$.
Step 3: Final Answer:
$r = 7$.