The ratio of the coefficient of x^15 to the term independent of x in the expansion of (x^2+ 2 x )^15 is:

$$\begin{aligned} & \mathrm{T}_{r+1}={ }^{15} \mathrm{C}_r\left(x^2\right)^{15-r} \cdot\left(2 x^{-1}\right)^r \\ & ={ }^{15} \mathrm{C}_r \times(2)^r \times x^{30-3 r} \end{aligned}$$​Tr+1​=15Cr​(x2)15−r⋅(2x−1)r=15Cr​×(2)r×x30−3r​ For independent term, $30-3 r=0 \Rightarrow r=10$30−3r=0⇒r=10 Hence the term independent of $x$x, $$\mathrm{T}_{11}={ }^{15} \mathrm{C}_{10} \times(2)^{10}$$T11​=15C10​×(2)10 For term involve $x^{15}, 30-3 r=15 \Rightarrow r=5$x15,30−3r=15⇒r=5 Hence coefficient of $x^{15}={ }^{15} \mathrm{C}_5 \times(2)^5$x15=15C5​×(2)5 Required ratio $$\begin{aligned} & =\frac{{ }^{15} \mathrm{C}_5 \times(2)^5}{{ }^{15} \mathrm{C}_{10} \times(2)^{10}}=\frac{\frac{15 !}{10 ! 5 !}}{\frac{15 !}{5 ! 10 !} \times(2)^5} \\ & =1: 32 \\ & \end{aligned}$$​=15C10​×(2)1015C5​×(2)5​=5!10!15!​×(2)510!5!15!​​=1:32​