The term without x in the expansion of

Question:

The term without $x$ in the expansion of $\left(2 x-\frac{1}{2 x^{2}}\right)^{12}$ is

(a) 495

(b) −495

(c) −7920

(d) 7920

Solution:

(d) 7920

Suppose the $(\mathrm{r}+1)$ th term in the given expansion is independent of $x$.

Then, we have :

$T_{r+1}={ }^{12} C_{r}(2 x)^{12-r}\left(\frac{-1}{2 x^{2}}\right)^{r}$

$=(-1)^{r}{ }^{12} C_{r} \quad 2^{12-2 r} \quad x^{12-r-2 r}$

For this term to be independent of $x$, we must have:

$12-3 r=0$

$\Rightarrow r=4$

$\therefore$ Required term :

$(-1)^{4}{ }^{12} C_{4} 2^{12-8}$

$=\frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2} \times 16$

$=7920$

 

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