Find the coefficient of ${x^8}$ in the expansion of ${\left( {{x^2} - \frac{2}{x}} \right)^7}$.

each term is of the form $\left( {\begin{array}{*{20}{c}}  7 \\   r \end{array}} \right){({x^2})^{7 - r}}{\left( {\frac{{ - 2}}{x}} \right)^\prime }$     (M1)

$= \left( {\begin{array}{*{20}{c}}  7 \\   r \end{array}} \right){x^{14 - 2r}}{( - 2)^r}{x^{ - r}}$

so $14 - 3r = 8$     (A1)

$r = 2$

so require $\left( {\begin{array}{*{20}{c}}  7 \\   2 \end{array}} \right){({x^2})^{}}^5{\left( {\frac{{ - 2}}{x}} \right)^2}$ (or simply $\left( {\begin{array}{*{20}{c}}  7 \\   2 \end{array}} \right){( - 2)^2}$)     A1

$= 21 \times 4$

$= 84$     A1

Note:     Candidates who attempt a full expansion, including the correct term, may only be awarded M1A0A0A0.

[4 marks]