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., $(r+1)^{t h}$ 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^{36-3 r-2 r}(-1)^r 2^r$
$ ={ }^{12} C_r(-1)^r 2^r x^{36-5 r}$
Now for this to contain $x^{11}$, we observe that
$36-5 r=11 \text {, i.e. } r=5$
Thus, the coefficient of $x^{11}$ is
${ }^{12} C_5(-1)^5 2^5=-\frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2} \times 32=-25344$