Question

The coefficient of a^–6 b⁴ in the expansion ${\left(\frac{1}{a}-\frac{2}{3}b\right)}^{10}$ is ___________.

Answer 1

$$\begin{gathered} \mathrm{In}{\left(\frac{1}{a}-\frac{2}{3}b\right)}^{10} \\ {T}_{r+1}{=}^{10}{C}_r{\left(\frac{1}{a}\right)}^{10-r}{\left(\frac{-2b}{3}\right)}^r \\ \mathrm{for}\mathrm{coefficient}\mathrm{of}{a}^{-6}{b}^4 \\ \mathrm{put}10-r=6 \\ \mathrm{i}.\mathrm{e}.r=4 \end{gathered}$$
We get
$\begin{array}{rcl}\mathrm{coefficient}\mathrm{of}{a}^{-6}{b}^4& =& {}^{10}{C}_4{\left(\frac{-2}{3}\right)}^4\\ & =& \frac{10!}{4!6!}\times \frac{2^4}{3^4}\\ & =& \frac{10\times 9\times 8\times 7}{2\times 3\times 4}\times \frac{2^4}{3^4}\\ & =& \frac{10\times 7\times 2^4}{3^3}\end{array}$

$\begin{array}{rcl}& & \mathrm{i}\end{array}\begin{array}{rcl}& & .\end{array}\begin{array}{rcl}& & \mathrm{e}\end{array}\begin{array}{rcl}& & .\end{array}\begin{array}{rcl}& & \end{array}\begin{array}{rcl}& & \mathrm{coefficient}\end{array}\begin{array}{rcl}& & \end{array}\begin{array}{rcl}& & \mathrm{of}\end{array}\begin{array}{rcl}& & \end{array}\begin{array}{rcl}& & a\end{array}\begin{array}{rcl}& & {}^{-6}\end{array}\begin{array}{rcl}& & \end{array}\begin{array}{rcl}& & b\end{array}\begin{array}{rcl}& & {}^4\end{array}\begin{array}{rcl}& & \end{array}\begin{array}{rcl}& =& \end{array}\begin{array}{rcl}& & \frac{1120}{27}\end{array}\begin{array}{rcl}& & .\end{array}$