Question:

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

Updated On: Sep 24, 2024

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Correct Answer: 5040

The correct answer is 5040.

(\frac{4 x}{5} + \frac{5}{2 x ^{2}})^{9}

Now,

T_{r + 1} =^{9} C_{r} \cdot (\frac{4 x}{5})^{9 - r} (\frac{5}{2 x ^{2}})^{r}

=^{9} C_{r} \cdot (\frac{4}{5})^{9 - r} (\frac{5}{2})^{r} \cdot x^{9 - 3 r}

Coefficient of

x^{- 6}

i.e.

9 - 3 r = - 6 \Rightarrow r = 5

So, Coefficient of

x^{- 6} =^{9} C_{5} (\frac{4}{5})^{4} \cdot (\frac{5}{2})^{5} = 5040

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