The term independent of x in the expansion of $\left(1-\frac{1}{2x}+3x^7\right){\left(x^2+\frac{1}{3x}\right)}^{10}$ is

(r + 1)th term in the expansion of ${\left(x^2+\frac{1}{3x}\right)}^{10}$is

$\begin{array}{l}{t}_{r+1}{=}^{10}{C}_r{\left(x^2\right)}^{10-r}{\left(\frac{1}{3x}\right)}^r\\ {=}^{10}{C}_r{\left(\frac{1}{3}\right)}^r{x}^{20-3r}\end{array}$

We require terms containing $x^0,x$ and $x^{-7}$ in the expansion of  ${\left(x^2+\frac{1}{3x}\right)}^{10}$

For $x^0$, set $20-3r=0$. No value of r.

For, $x^1$ set $20-3r=1$. No value of r.

For, $x^{-7}$ set $20-3r=-7\Rightarrow r=9$.

Thus, required coefficient is

$3\left[{ }^{10}{C}_9{\left(\frac{1}{3}\right)}^9\right]=\frac{10}{3^8}$