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  • The coefficient of , in the expansion of <img alt="\mathrm{\left(\frac{4 x}{5}+\frac{5

The coefficient of \mathrm{x^{-6}}, in the expansion of \mathrm{\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9}, is
 

Option: 1

5040


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\mathrm{ \left(\frac{4 x}{5} \times \frac{5}{2 x^2}\right)^9 }
General term is
\mathrm{ \begin{aligned} & \mathrm{T}_{\mathrm{r}+1}={ }^9 \mathrm{C}_r\left(\frac{4 \mathrm{n}}{\mathrm{r}}\right)^{9-r}\left(\frac{5}{2 \mathrm{n}^2}\right)^r \\ & ={ }^9 \mathrm{C}_r\left(\frac{4}{5}\right)^{9-r}\left(\frac{5}{2}\right)^r x^{9-\mathrm{r}-2 \mathrm{r}} \end{aligned} }
For coefficient of term \mathrm{\mathrm{x}^{-6}}   \begin{aligned} & 9-\mathrm{r}-2 \mathrm{r}=-6 \\ \end{aligned}

                                          \begin{aligned} & 15=3 \mathrm{r} \end{aligned}

                                          \mathrm{\mathrm{ r=5 }}

\mathrm{ \text { Coefficient of term x }{ }^{-6} \: \: \: \: \mathrm{^{9}C_{5}\left ( \frac{4}{5} \right )^{4}\left ( \frac{5}{2} \right )^{5}} }
                                            \mathrm{=\frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} \times 5 \times 2^3 }
                                            \mathrm{ =9 \times 2 \times 7 \times 5 \times 8 }
                                            \mathrm{ =5040}
 

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Rishabh

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