Question

If $x^r$ occurs in the expansion of $ \left( x + \frac{1}{x} \right)^n $, then its coefficient is

• A. $ \frac{n!}{(r!)^2}$
• B. $ \frac{n!}{(r +1)! (r-1)!}$
• C. $\frac{n!}{\left(\frac{n+r}{2}\right)!\left(\frac{n-r}{2}\right)!}$
• D. $\frac{n!}{\left[\left(\frac{r}{2}\right)!\right]^{2}}$

Answer 1

Answer: (C)

$T_{m+1}=\,{}^{n}C_{m }\left(x\right)^{n-m}\left(\frac{1}{x}\right)^{m}$
$ =\,{}^{n}C_{m} x^{n-2m}$
$n-3m = r \Rightarrow m = \frac{n-r}{2}$
Coefficient of $ x^{r} =\,{}^{n}C_{\left(\frac{n-r}{2}\right)}=\frac{n!}{\left(\frac{n-r}{2}\right)!\left(\frac{n+r}{2}\right)!}$