The term independent of $x$ in ${(2 x^{1 / 2} - 3 x^{- 1 / 3})}^{20}$ is

^{20} C_{8} {.6}^{8} {.2}^{4}

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^{20} C_{8} {.2}^{8} {.3}^{8}

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^{20} C_{8} {.6}^{8} {.3}^{4}

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^{20} C_{12} {.6}^{12}

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Solution

The correct option is C $^{20} C_{8} {.6}^{8} {.3}^{4}$
$T_{r + 1} =^{n} C_{r} a^{n - r} b^{r}$
Applying to the above question, we get
$T_{r + 1} = (- 1)^{r}^{20} C_{r} 2^{20 - r} 3^{r} x^{10 - \frac{r}{2} - \frac{r}{3}}$ ...(i)
For term independent of $x$
$\Rightarrow 10 - \frac{r}{2} - \frac{r}{3} = 0$
$\Rightarrow 10 = \frac{5 r}{6}$
$\Rightarrow r = 12$
Substituting in equation (i), we get
$T_{13} =^{20} C_{12} 2^{8} 3^{12}$
$=^{20} C_{12} 6^{8} 3^{4}$
Now $^{n} C_{r} =^{n} C_{n - r}$
Therefore
$^{20} C_{12} 6^{8} 3^{4} =^{20} C_{20 - 12} 6^{8} 3^{4}$
$=^{20} C_{8} 6^{8} 3^{4}$

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