${ }^{\mathrm{n}} \mathrm{C}_1 \mathrm{x}^{\mathrm{n}-1} \mathrm{y}=135$....(i)
${ }^n C_2 x^{n-2} y^2=30$....(ii)
${ }^{\mathrm{n}} \mathrm{C}_3 \mathrm{x}^{\mathrm{n}-3} \mathrm{y}^3=\frac{10}{3}$....(iii)
$\begin{aligned} & \text { By } \frac{(i)}{(\text { ii) }} \\ & \frac{{ }^n C_1}{{ }^n C_2} \frac{x}{y}=\frac{9}{2}...(iv) \\ & \text { By } \frac{(\text { ii) }}{(\text { iii) }} \\ & \frac{{ }^n C_2}{{ }^n C_3} \frac{x}{y}=9....(v)\end{aligned}$
$\begin{aligned} & \text { By } \frac{(\text { iv })}{(v)} \\ & \frac{{ }^n C_1{ }^n C_3}{{ }^n C_2{ }^n C_2}=\frac{1}{2} \\ & \frac{2 n^2(n-1)(n-2)}{6}=\frac{n(n-1)}{2} \frac{n(n-1)}{2} \\ & 4 n-8=3 n-3 \\ & \Rightarrow n=5 \\ & \text { put in (v) }\end{aligned}$
$\begin{aligned} & \frac{x}{y}=9 \\ & x=9 y \\ & \text { put in (i) } \\ & { }^5 C_1 x^4\left(\frac{x}{9}\right)=135 \\ & x^5=27 \times 9 \\ & \Rightarrow x=3, \quad y=\frac{1}{3} \\ & 6\left(n^3+x^2+y\right) \\ & =6\left(125+9+\frac{1}{3}\right) \\ & =806\end{aligned}$