Explanation

(c)
The general term in the expansion of $\left( \frac{3}{2}x^{2} - \frac{1}{3x} \right)^{9}$is
$T_{r + 1} = \ ^{9}C_{r}\left( \frac{3}{2}x^{2} \right)^{9 - r}\left( - \frac{1}{3x} \right)^{r}$
$= \ ^{9}C_{r}\left( \frac{3}{2} \right)^{9 - r}\left( - \frac{1}{3} \right)^{r}x^{18 - 3r}$ …(i)
Now, the coefficients of the terms $x^{0\ },x^{- 1}$ and $x^{- 3}$ in $\left( \frac{3}{2}x^{2} - \frac{1}{3x} \right)^{9}$is
For $x^{0},\ 18 - 3r = 0\ \Rightarrow \ r = 6$
For $x^{- 1}$, there exists no integer value of $r$
For $x^{- 3}$, $18 - 3r = - 3\ \Rightarrow \ r = 7$
Now, the coefficient of the term independent of $x$ in the expansion of $\left( 1 + x + 2x^{3} \right)\left( \frac{3}{2}x^{2} - \frac{1}{3x} \right)^{9}$
$= 1.\ ^{9}C_{6}( - 1)^{6}\left( \frac{3}{2} \right)^{9 - 6}\left( \frac{1}{3} \right)^{6} + 0 + 2.\ ^{9}C_{7}{( - 1)}^{7}\left( \frac{3}{2} \right)^{9 - 7}\left( \frac{1}{3} \right)^{7}$
$= \frac{9.8.7}{1.2.3}.\frac{3^{3}}{2^{3}}.\frac{1}{3^{6}} + 2.\frac{9.8}{1.2}( - 1)\frac{3^{2}}{2^{2}}.\frac{1}{3^{7}}$
$= \frac{7}{18} - \frac{2}{27} = \frac{17}{54}$