(x3+2y)4= 4C0(x3)4+ 4C1(x3)3(2y)+ 4C2(x3)2(2y)2+ 4C3(x3)(2y)3+ 4C4(2y)4=x481+8x327y+8x23y2+32x3y3+16y4

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${(\frac{x}{3} + \frac{2}{y})}^{4} =$

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$\frac{x^{4}}{80} + \frac{8 x^{3}}{25 y} + \frac{8 x^{2}}{3 y^{2}} + \frac{16}{2 y^{4}}$

$\frac{x^{4}}{80} + \frac{8 x^{3}}{27 y} + \frac{8 x^{2}}{3 y^{2}} + \frac{32 x}{3 y^{3}} + 16$

$\frac{x^{4}}{81} + \frac{8 x^{3}}{27 y} + \frac{8 x^{2}}{3 y^{2}} + \frac{32 x}{3 y^{3}} + \frac{16}{y^{4}}$

$\frac{x^{4}}{80} + \frac{8 x^{3}}{27 y} + \frac{8 x^{2}}{3 y^{2}} + \frac{32 x}{3 y^{3}} + 16$

answer is C.

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Detailed Solution

${(\frac{x}{3} + \frac{2}{y})}^{4} =^{4} C_{0} {(\frac{x}{3})}^{4} +^{4} C_{1} {(\frac{x}{3})}^{3} (\frac{2}{y}) +^{4} C_{2} {(\frac{x}{3})}^{2} {(\frac{2}{y})}^{2} +^{4} C_{3} (\frac{x}{3}) {(\frac{2}{y})}^{3} +^{4} C_{4} {(\frac{2}{y})}^{4}$

$= \frac{x^{4}}{81} + \frac{8 x^{3}}{27 y} + \frac{8 x^{2}}{3 y^{2}} + \frac{32 x}{3 y^{3}} + \frac{16}{y^{4}}$

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