The coefficient of x^-6 in the expansion of (4x5+52x^2)^9 is _______ Answer: 5040

Question

The coefficient of $x^{-6}$ in the expansion of $\left(\dfrac{4x}{5}+\dfrac{5}{2x^2}\right)^9$ is \_\_\_\_\_\_\_

Answer: 5040

Accepted Answer

$$T_{r+1}=\binom{9}{r}\left(\frac{4x}{5}\right)^{9-r}\left(\frac{5}{2x^2}\right)^r=\binom9r\left(\frac45\right)^{9-r}\left(\frac52\right)^rx^{9-3r}$$

For $x^{-6}$: $9-3r=-6\Rightarrow r=5$.

$$\binom95\left(\frac45\right)^4\left(\frac52\right)^5=126\times\frac{256}{625}\times\frac{3125}{32}=5040$$

Solutions · 1