Q153 / 270 · NDA 2018 (Phase 1) · Mathematics

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

  • A
    5
  • Correct answer
    B
    14
  • C
    21
  • D
    22

Solution

Given, in the expansion $\mathrm{g}(1+\mathrm{x})^{43}$, coefficients of $(2 \mathrm{r}+1)^{\mathrm{th}}$ term and $(\mathrm{r}+2)^{\mathrm{th}}$ term are equal. Coefficient of $(2 \mathrm{r}+1)^{\text {th }}$ term $={ }^{\mathrm{n}} \mathrm{C}_{2 \mathrm{r}}$ Coefficient of $(\mathrm{r}+2)^{\text {th }}$ term $=\mathrm{n}_{\mathrm{C}_{\mathrm{r}+1}}$ $\begin{aligned} & \mathrm{n}_{\mathrm{C}_{2 \mathrm{r}}}=\mathrm{n}_{\mathrm{C}_{\mathrm{r}+1}} \\ & \Rightarrow 43_{\mathrm{C}_{2 \mathrm{r}}}=43_{\mathrm{C}_{\mathrm{r}+1}} (\because \mathrm{n}=43)\\ & \Rightarrow 2 \mathrm{r}+\mathrm{r}+1=43 \\ & \Rightarrow 3 \mathrm{r}+1=43 \\ & \Rightarrow 3 \mathrm{r}+42 \Rightarrow \mathrm{r}=14 \end{aligned}$

Hence, the correct answer is option: 'B'.

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