The coefficient of x^7 in the expansion of le | vedclass

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(C) The general term in the expansion of $(a + b)^n$ is given by $T_{r+1} = {}^nC_r a^{n-r} b^r$.Here,$a = \frac{x^2}{2}$,$b = -\frac{2}{x}$,and $n =

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$-14$

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(C) The general term in the expansion of $(a + b)^n$ is given by $T_{r+1} = {}^nC_r a^{n-r} b^r$.Here,$a = \frac{x^2}{2}$,$b = -\frac{2}{x}$,and $n = 8$.$T_{r+1} = {}^8C_r \left( \frac{x^2}{2} ight)^{8-r} \left( -\frac{2}{x} ight)^r$$T_{r+1} = {}^8C_r \left( \frac{1}{2} ight)^{8-r} (x^2)^{8-r} (-2)^r (x^{-1})^r$$T_{r+1} = {}^8C_r \left( \frac{1}{2} ight)^{8-r} (-2)^r x^{16-2r} x^{-r}$$T_{r+1} = {}^8C_r \left( \frac{1}{2} ight)^{8-r} (-2)^r x^{16-3r}$For the coefficient of $x^7$,we set $16 - 3r = 7$,which gives $3r = 9$,so $r = 3$.The coefficient is ${}^8C_3 \left( \frac{1}{2} ight)^{8-3} (-2)^3 = {}^8C_3 \left( \frac{1}{2} ight)^5 (-8)$.$= \frac{8 	imes 7 	imes 6}{3 	imes 2 	imes 1} 	imes \frac{1}{32} 	imes (-8) = 56 	imes \left( -\frac{8}{32} ight) = 56 	imes \left( -\frac{1}{4} ight) = -14$.

The coefficient of $x^7$ in the expansion of $\left( \frac{x^2}{2} - \frac{2}{x} \right)^8$ is

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