Question

Find the coefficient of $x^{11}$ in the expansion of ${\left(x^3-\frac{2}{x^2}\right)}^{12}$.

• A. $-25344$
• B. $-25250$
• C. $-25000$
• D. $24310$

Answer 1

Answer: (A)

Let $(r+1{)}^{th}$ term contain $x^{11}$.
We have, $T_{r+1}={}^{12}{C}_r{\left(x^3\right)}^{12-r}{\left(-\frac{2}{x^2}\right)}^r$
$={}^{12}{C}_r{x}^{36-3r-2r}{\left(-1\right)}^r{2}^r$
$={}^{12}{C}_r{\left(-1\right)}^r{2}^r{x}^{36-5r}$
Now for this to contain $x^{11}$, $36-5r=11$, i.e., $r=5$
Thus, the coefficient of $x^{11}$ is ${}^{12}{C}_5{\left(-1\right)}^5{2}^5=-25344$