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Consider the expansion $\left(x^{2}+\frac{1}{x}\right)^{15}$.
What is the ratio of coefficient of $\mathrm{x}^{15}$ to the term independent of $\mathrm{x}$ in the given expansion?
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What is the ratio of coefficient of $\mathrm{x}^{15}$ to the term independent of $\mathrm{x}$ in the given expansion?
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The correct answer is:
1
For coefficient of $\mathrm{x}^{15}$,
$30-3 r=15$
$\Rightarrow \mathrm{r}=5$
$\therefore$ the coefficient of $\mathrm{x}^{15} \mathrm{is}^{15} \mathrm{C}_{5}$
and coefficient of independent of $\mathrm{x}$ is
$30-3 \mathrm{r}=0$
$\Rightarrow \mathrm{r}=10$
So, coefficient of independent of $\mathrm{x}$ is ${ }^{15} \mathrm{C}_{10}$.
$\therefore$ Required ratio $=\frac{15 \mathrm{C}_{5}}{{ }^{15} \mathrm{C}_{10}}=\frac{{ }^{15} \mathrm{C}_{5}}{{ }^{15} \mathrm{C}_{5}}=1$
$\left(\because{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}-\mathrm{r}}\right)$
$30-3 r=15$
$\Rightarrow \mathrm{r}=5$
$\therefore$ the coefficient of $\mathrm{x}^{15} \mathrm{is}^{15} \mathrm{C}_{5}$
and coefficient of independent of $\mathrm{x}$ is
$30-3 \mathrm{r}=0$
$\Rightarrow \mathrm{r}=10$
So, coefficient of independent of $\mathrm{x}$ is ${ }^{15} \mathrm{C}_{10}$.
$\therefore$ Required ratio $=\frac{15 \mathrm{C}_{5}}{{ }^{15} \mathrm{C}_{10}}=\frac{{ }^{15} \mathrm{C}_{5}}{{ }^{15} \mathrm{C}_{5}}=1$
$\left(\because{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}-\mathrm{r}}\right)$
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