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In the expansion of $(1+\mathrm{x})^{43}$, if the coefficients of $(2 \mathrm{r}+1)^{\text {th }}$ and $(\mathrm{r}+2)^{\text {th }}$ terms are equal, then what is the value of $\mathrm{r}(\mathrm{r} \neq 1)$ ?
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The correct answer is:
14
Given, in the expansion $\mathrm{g}(1+\mathrm{x})^{43}$, coefficients of $(2 \mathrm{r}+1)^{\text {th }}$ 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 $=\mathrm{n}_{\mathrm{C}_{\mathrm{r}+1}}$
$\mathrm{n}_{\mathrm{C}_{2 \mathrm{r}}}=\mathrm{n}_{\mathrm{C}_{\mathrm{r}+1}}$
$\Rightarrow 43_{\mathrm{C}_{2 \mathrm{r}}}=43_{\mathrm{C}_{\mathrm{r}+1}}$
$(\because \mathrm{n}=43)$
$\Rightarrow 2 \mathrm{r}+\mathrm{r}+1=43$
$\Rightarrow 3 r+1=43$
$\Rightarrow 3 \mathrm{r}+42 \Rightarrow \mathrm{r}=14$
Coefficient of $(2 \mathrm{r}+1)^{\text {th }}$ term $=\mathrm{n}_{\mathrm{C}_{2 \mathrm{r}}}$
Coefficient of $(\mathrm{r}+2)^{\text {th }}$ term $=\mathrm{n}_{\mathrm{C}_{\mathrm{r}+1}}$
$\mathrm{n}_{\mathrm{C}_{2 \mathrm{r}}}=\mathrm{n}_{\mathrm{C}_{\mathrm{r}+1}}$
$\Rightarrow 43_{\mathrm{C}_{2 \mathrm{r}}}=43_{\mathrm{C}_{\mathrm{r}+1}}$
$(\because \mathrm{n}=43)$
$\Rightarrow 2 \mathrm{r}+\mathrm{r}+1=43$
$\Rightarrow 3 r+1=43$
$\Rightarrow 3 \mathrm{r}+42 \Rightarrow \mathrm{r}=14$
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