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(D) The general term $T_{r+1}$ in the expansion of $(y^{-1/6} - y^{1/3})^9$ is given by $T_{r+1} = ^{9}C_{r} (y^{-1/6})^{9-r} (-y^{1/3})^r$.For the te

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$-84$

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(D) The general term $T_{r+1}$ in the expansion of $(y^{-1/6} - y^{1/3})^9$ is given by $T_{r+1} = ^{9}C_{r} (y^{-1/6})^{9-r} (-y^{1/3})^r$.For the term to be independent of $y$,the exponent of $y$ must be $0$.So,$-\frac{1}{6}(9-r) + \frac{1}{3}r = 0$.Multiplying by $6$,we get $-(9-r) + 2r = 0$,which simplifies to $-9 + r + 2r = 0$,so $3r = 9$,which means $r = 3$.The term independent of $y$ is $T_{3+1} = ^{9}C_{3} (y^{-1/6})^{6} (-y^{1/3})^3$.Calculating the value: $^{9}C_{3} = \frac{9 	imes 8 	imes 7}{3 	imes 2 	imes 1} = 84$.Thus,$T_{4} = 84 	imes (y^{-1}) 	imes (-y) = -84$.

The term independent of $y$ in the expansion of $(y^{-1/6} - y^{1/3})^9$ is

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