Question

The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of ${(x^2+\frac{2}{x})}^{15}$ is

• A. $1:8$
• B. $1:12$
• C. $1:16$
• D. $1:32$

Answer 1

The correct option is D $1:32$

Writing the general term, we get
$T_{r+1}={}^{15}{C}_r{x}^{30-3r}{2}^r$
Hence for the term independent of x, $30-3r=0$
$r=10$
Coefficient will be ${}^{15}{C}_{10}{2}^{10}$
$=3003\left(2^{10}\right)$ ...(i)
The term with $x^{15}$ will be $30-3r=15$
$r=5$
Coefficient will be ${}^{15}{C}_5{2}^5$
$=3003\left(2^5\right)$ ...(ii)
Therefore $\frac{ii}{i}$
$=\frac{2^5}{2^{10}}$
$=\frac{1}{2^5}$
$=1:32$