Given that the term of the expansion (x^{1/3} | vedclass

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(C) The general term $T_{r + 1}$ in the expansion of $(x^{1/3} - x^{-1/2})^{15}$ is given by:$T_{r + 1} = ^{15}C_r (x^{1/3})^{15 - r} (-x^{-1/2})^r$$T

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$1001$

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(C) The general term $T_{r + 1}$ in the expansion of $(x^{1/3} - x^{-1/2})^{15}$ is given by:$T_{r + 1} = ^{15}C_r (x^{1/3})^{15 - r} (-x^{-1/2})^r$$T_{r + 1} = ^{15}C_r (-1)^r x^{(15 - r)/3} x^{-r/2}$$T_{r + 1} = ^{15}C_r (-1)^r x^{(15 - r)/3 - r/2}$For the term to be independent of $x$,the exponent of $x$ must be $0$:$\frac{15 - r}{3} - \frac{r}{2} = 0$$2(15 - r) - 3r = 0$$30 - 2r - 3r = 0$$5r = 30 \Rightarrow r = 6$Substituting $r = 6$ into the general term:$T_7 = ^{15}C_6 (-1)^6 = ^{15}C_6 = \frac{15 	imes 14 	imes 13 	imes 12 	imes 11 	imes 10}{6 	imes 5 	imes 4 	imes 3 	imes 2 	imes 1} = 5005$Given $T_7 = 5m$,we have $5005 = 5m \Rightarrow m = 1001$.

Given that the term of the expansion $(x^{1/3} - x^{-1/2})^{15}$ which does not contain $x$ is $5m$,where $m \in N$,then $m =$

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