Find the term independent of


$\mathrm{x}, \mathrm{x} \neq 0$, in the expansion of $\left(\frac{3 x^{2}}{2}-\frac{1}{3 x}\right)^{15}$.


Given $\left(\frac{3 x^{2}}{2}-\frac{1}{3 x}\right)^{15}$

From the standard formula of $T_{r+1}$ we can write given expression as

$T_{r+1}={ }^{15} C_{r}\left(\frac{3 x^{2}}{2}\right)^{15-r}\left(-\frac{1}{3 x}\right)^{r}$

$T_{r+1}={ }^{15} C_{r}(-1)^{r} 3^{15-2 r} 2^{r-15} x^{30-3 r}$

For the term independent of $x$, we have

$30-3 r=0$

Which implies $r=10$

By substituting the value of $r$ in above obtained expression we get

$T_{10+1}={ }^{15} C_{10} 3^{-5} 2^{-5}$

$={ }^{15} C_{10}\left(\frac{1}{6}\right)^{5}$

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