Question

In the second term in the expansion ${(\sqrt[13]{13a}+\frac{a}{\sqrt{a^{-1}}})}^n$ is $14a^{\frac{5}{2}}$ , then the value of $\frac{{}^n{C}_3}{{}^n{C}_2}$ is

• A. $8$
• B. $12$
• C. $4$
• D. None of these

Answer 1

Answer: (C)

$4$

Given: $T_2=14a^{\frac{5}{2}}$

$\Rightarrow {{}^{n}C}_1{\left(a^{\frac{1}{13}}\right)}^{n-1}.{\left(\frac{a}{a^{-\frac{1}{2}}}\right)}^1=14a^{\frac{5}{2}}$

$\Rightarrow n.a^{\frac{n-1}{13}}.a^{\frac{3}{2}}=14a^{\frac{5}{2}}$

$\Rightarrow n.a^{\frac{n-1}{13}}=14a\Rightarrow n.a^{\frac{n-14}{13}}=14\Rightarrow n=14$

$\therefore \frac{{{}^{n}C}_3}{{{}^{n}C}_2}=\frac{{{}^{14}C}_3}{{{}^{14}C}_2}=4$