Question

If the fifth term in the expansion of ${(x^{1/3}+\frac{1}{x})}^n$
does not depend upon $x$, then ${{}^{n}C}_2+{{}^{n}C}_{14}$ equals

• A. $240$
• B. $480$
• C. $360$
• D. $120$

Answer 1

The correct option is A $240$

Given expansion is ${(x^{\frac{1}{3}}+\frac{1}{x})}^n$

The general term is ${}^n{C}_r{x}^{\frac{n-r}{3}}{x}^{-r}$

Given fifth term is independent of $x$,

$⟹r=4$,

Subtituting the value of $r$ in the general term gives,

$t_5={}^n{C}_4{x}^{\frac{n-16}{3}}$

Given that this value is independent of $x$

$⟹n=16$

$\therefore$ the required value is ${{16}_C}_2+{{16}_C}_{14}⟹2\left({}^{16}{C}_2\right)⟹240$

formula used is ${n_C}_r={n_C}_{n-r}$