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  4. If the second term in the expansion (√[13]a+(a/√a-1))n is 14a5/2, then (nC3/nC2)=

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Q. If the second term in the expansion $\left(\sqrt[13]{a}+\frac{a}{\sqrt{a^{-1}}}\right)^{n}$ is $14a^{5/2}$, then $\frac{^{n}C_{3}}{^{n}C_{2}}=$

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Solution:

We have $T_{2}=14\,a^{\frac{5}{2}}$
$\Rightarrow ^{n}C_{1}\left(a^{\frac{1}{13}}\right)^{n-1} \left(a^{\frac{3}{2}}\right)=14a^{\frac{5}{2}}$
$na^{\frac{n-1}{13}+\frac{3}{2}}=14a^{\frac{5}{2}} \Rightarrow n=14$
$\Rightarrow \frac{^{n}C_{3}}{^{n}C_{2}}=\frac{^{14}C_{3}}{^{14}C_{2}}=\frac{12}{3}=4$