Question

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

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

Answer 1

Answer: (B)

Let $T_{r+1}$ be the general term of ${\left(x^2+\frac{2}{x}\right)}^{15}$
$\therefore T_{r+1}={}^{15}{C}_r{\left(x^2\right)}^{15-r}{\left(\frac{2}{x}\right)}^r$
$={}^{15}{C}_r{\left(2\right)}^r{x}^{30-3r}\dots \left(1\right)$
Now, for the coefficient of term containing $x^{15}$,
$30-3r=15$, i.e., $r=5$
Therefore, ${}^{15}{C}_5{\left(2\right)}^5$ is the coefficient of $x^{15}$ (from $\left(1\right)$)To find the term independent of $x$, put $30-3r=0$
$\Rightarrow r=10$ Thus ${}^{15}{C}_{10}{2}^{10}$ is the term independent of $x$ (from $\left(1\right)$)
Now the ratio is $\frac{{}^{15}{C}_5{2}^5}{{}^{15}{C}_{10}{2}^{10}}=\frac{1}{2^5}=\frac{1}{32}$