Find the term independent of $x$ in the expansion of ${(2 x^{1 / 2} - \frac{x^{- 1 / 3}}{3})}^{20}$

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Solution

$T_{r + 1} = {^{20} C}_{r} {(2 x)}^{((20 - r) / 2)} . {(\frac{{- x}^{1 / 3}}{3})}^{r}$
$\Rightarrow \frac{20 - r}{2} + \frac{- r}{3} = 0$
$\Rightarrow r = 12$
$∴$ Coefficient of constant term,
$= {^{20} C}_{12} . 2^{4} . {(\frac{- 1}{3})}^{12}$
$= {^{20} C}_{12} . \frac{2^{4}}{3^{12}} .$

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