The position of the term independent of x in the expansion of
Question:

The position of the term independent of $x$ in the expansion of $\left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^{2}}\right)^{10}$ is _____________

Solution:

for $\left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^{2}}\right)^{10}$

$T_{r+1}={ }^{10} C_{r}\left(\sqrt{\frac{x}{3}}\right)^{10-r}\left(\frac{3}{2 x^{2}}\right)^{r}$

$={ }^{10} C_{r} \frac{1}{(\sqrt{3})^{10-r}} x^{\frac{10-r}{2}} x^{-2 r}\left(\frac{3}{2}\right)^{r}$

$T_{r+1}=\frac{{ }^{10} C_{r}}{(\sqrt{3})^{10-r}}\left(\frac{3}{2}\right)^{r} \quad x^{\frac{10-5 r}{2}}$

for constant term, $\frac{10-5 r}{2}=0 \quad$ i. e. $r=2$

Hence, third term is independent of x.

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