The coefficient of the term independent of x in the expansion of

Question:

The coefficient of the term independent of $x$ in the expansion of $\left(a x+\frac{b}{x}\right)^{14}$ is

(a) $14 ! a^{7} b^{7}$

(b) $\frac{14 !}{7 !} a^{7} b^{7}$

(c) $\frac{14 !}{(7 !)^{2}} a^{7} b^{7}$

(d) $\frac{14 !}{(7 !)^{3}} a^{7} b^{7}$

Solution:

(c) $\frac{14 !}{(7 !)^{2}} a^{7} b^{7}$

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

Then, we have

$T_{r+1}={ }^{14} C_{r}(a x)^{14-r}\left(\frac{b}{x}\right)^{r}$

$={ }^{14} C_{r} a^{14-r} b^{r} x^{14-2 r}$

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

$14-2 r=0$

$\Rightarrow r=7$

$\therefore$ Required term $={ }^{14} C_{7} a^{14-7} b^{7}=\frac{14 !}{(7 !)^{2}} a^{7} b^{7}$