Question

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

• A. $14!a^7{b}^7$
• B. $\frac{14!}{7!}{a}^7{b}^7$
• C. $\frac{14!}{(7!{)}^2}$
• D. $\frac{14!}{(7!{)}^3}{a}^7{b}^7$

Answer 1

The correct option is C

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

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

Then, we have:

$T_{r+1}={}^{14}{C}_r(ax{)}^{14-r}{\left(\frac{b}{x}\right)}^r$

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

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

14-2r=0

$\Rightarrow r=7$

$\therefore$ Required term $={}^{14}{C}_7{a}^{14-7}{b}^7=\frac{14!}{(7!)}{a}^7{b}^7$