Question

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

• A. 21344
• B. $-21344$
• C. 25344
• D. $-25344$

Answer 1

Answer: (D)

Let the general term, i.e., $(r+1{)}^{th}$ contain $x^{11}$.
We have, $T_{r+1}={}^{12}{C}_r{\left(x^3\right)}^{12-r}{\left(-\frac{2}{x^2}\right)}^r$
$={}^{12}{C}_r{x}^{36-3r-2r}(-1{)}^r{2}^r$
$={}^{12}{C}_r(-1{)}^r{2}^r{x}^{36-5r}$
Now for this to contain $x^{11}$, we observe that
$36-5r=11\text{, i.e. }r=5$
Thus, the coefficient of $x^{11}$ is
${}^{12}{C}_5(-1{)}^5{2}^5=-\frac{12\times 11\times 10\times 9\times 8}{5\times 4\times 3\times 2}\times 32=-25344$