By considering the general term in the expansion of $\left(\dfrac{2}{x^3}-x^2\right)^{10}$, explain why there is no term in $x^6$. Find the coefficient of $x^6$ in the expansion of $\left(\dfrac{2}{x^3}-x^2\right)^{10}\left(3-\dfrac{x^6}{8}\right).

1. Write down the general term of the expansion

The general term is given by:
$T_{r+1} = \binom{10}{r} \left(\frac{2}{x^3}\right)^{10-r} (-x^2)^r$

2. Simplify the general term to isolate the power of $x$

$T_{r+1} = \binom{10}{r} 2^{10-r} x^{-3(10-r)} (-1)^r x^{2r}$
$T_{r+1} = \binom{10}{r} 2^{10-r} (-1)^r x^{-30+3r+2r}$
$T_{r+1} = \binom{10}{r} 2^{10-r} (-1)^r x^{5r-30}$

3. Equate the power of $x$ to 6 and solve for $r$ to explain the absence of the term

For the term in $x^6$, we require the power of $x$ to be 6:
$5r - 30 = 6$
$5r = 36$
$r = 7.2$
Since $r$ must be an integer ($0 \leqslant r \leqslant 10$), and $7.2$ is not an integer, there is no term in $x^6$.

1. Identify the terms from the binomial expansion needed to form $x^6$ in the product.

To find the term in $x^6$ in the expansion of $\left(\frac{2}{x^3}-x^2\right)^{10}\left(3-\frac{x^6}{8}\right)$, we consider the product of terms.
We need:
(Term in $x^6$ from expansion) $\times 3$ $+$ (Term in $x^0$ from expansion) $\times \left(-\frac{x^6}{8}\right)$.
From part (a), there is no term in $x^6$ in the expansion of $\left(\frac{2}{x^3}-x^2\right)^{10}$. Thus, we only need to find the constant term ($x^0$).

2. Find the value of $r$ for the constant term in the binomial expansion.

The general term of $\left(\frac{2}{x^3}-x^2\right)^{10}$ is:
$T_{r+1} = \binom{10}{r} \left(\frac{2}{x^3}\right)^{10-r} (-x^2)^r = \binom{10}{r} 2^{10-r} (-1)^r x^{5r-30}$
For the constant term, the power of $x$ must be $0$:
$5r - 30 = 0 \implies r = 6$

3. Calculate the coefficient of the constant term.

Substitute $r = 6$ into the general term:
$T_7 = \binom{10}{6} 2^{10-6} (-1)^6 x^0$
$T_7 = 210 \times 2^4 \times 1 = 210 \times 16 = 3360$
So, the constant term is $3360$.

4. Calculate the final coefficient of $x^6$ in the product.

The term in $x^6$ in the product is:
$3360 \times \left(-\frac{x^6}{8}\right) = -420x^6$
Therefore, the coefficient of $x^6$ is $-420$.