Question

Find $A_2^n$, if the fifth term of the expansion of ${(\sqrt[3]{3x}+\frac{1}{x})}^n$ does not depend on $x$.

Answer 1

Given

${(\sqrt[3]{3x}+\frac{1}{x})}^n$

General term

$T_{r+1}={\sum }_{r=0}^n{}^n{C}_r(\sqrt[3]{3x}{)}^{n-r}{\left(\frac{1}{x}\right)}^r$

$T_{r+1}={\sum }_{r=0}^n{}^n{C}_r(x{)}^{\frac{n-r}{3}-r}$

$T_{r+1}={\sum }_{r=0}^n{}^n{C}_r(x{)}^{\frac{n-4r}{3}}$

Fifth term i.e. $r=4$ in given expansion is independent of x

Hence $\frac{n-4r}{3}=0\Rightarrow n=4\times 4=16$

Second term be

$T_{1+1}={}^{16}{C}_1(x{)}^{\frac{16-4}{3}}$

$T_2=16(x{)}^4$

$A_2^n=A_2^{16}={16}^{16}$