Find the term independent of


Find the term independent of $\mathrm{x}$ in the expansion of $\left(3 x-\frac{2}{x^{2}}\right)^{15}$


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

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

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

For the term independent of $x$, we have $15-3 r=0$

Which implies $r=5$

The term independent of $x$ is

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

$=\frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 !}{5 \times 4 \times 3 \times 2 \times 1 \times 10 !} \cdot 3^{10} \cdot 2^{5}$

$=-3003 \times 3^{10} \times 2^{5}$

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