If an the expansion of $(1+\text{x})^{15},$ the coefficients of $(2\text{r}+3)^{\text{th}}$ and $(\text{r}-1)^{\text{th}}$ terms are equal, then the value of r is:

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Aicharya · Accepted Answer

Solution: Coefficients of $(2\text{r}+3)^{\text{th}}$ and $(\text{r}-1)^{\text{th}}$ terms in the given expansion are ${^\text{15}}\text{C}_{\text{2r}+2}$ and ${^\text{15}}\text{C}_{\text{2r}-2}.$ Then, we have ${^\text{15}}\text{C}_{\text{2r}+2}={^\text{15}}\text{C}_{\text{r}-2}$ $\Rightarrow 2\text{r}+2=\text{r}-2$ or $2\text{r}+2+\text{r}-2=15$ $\Rightarrow \text{r}=-4$ or $\text{r}=5$ Neglecting the negative value, We have $\text{r}=5$

If an the expansion of \((1+\text{x})^{15},\) the coefficients of \((2\text{r}+3)^{\text{th}}\) and…

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