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

$\begin{array}{1 1}(A)\;-3003(3^{10})(2^5)\\(B)\;3003(3^{10})(2^5)\\(C)\;300(3^{11})(2^5)\\(D)\;-300(3^{15})(2^{10})\end{array}$

Toolbox:
• $T_{r+1}=nC_ra^{n-r}b^r$
$(3x-\large\frac{2}{x^2})^{15}$$=(-1)^r 15C_r(3x)^{n-r}(\large\frac{2}{x^2})^r T_{r+1}=(-1)^r 15C_r(3)^{15-r}(x)^{n-r}\large\frac{2^r}{x^{2r}} \Rightarrow (-1)^r 15C_r(3)^{15-r}(x)^{15-r}\large\frac{2^r}{x^{2r}} Since the term is independent of x in the expansion we have 15-r-2r=0 15-3r=0 r=5 Hence 6^{th} term is independent of x T_6=(-1)^515C_5(3)^{15-5}.2^5 T_6=(-1)^515C_5 3^{10}.2^5 \Rightarrow \large\frac{15!}{5!10!}$$ 3^{10}.2^5$