Question

The middle term in the expansion of ${(x+\frac{1}{x})}^{10}$ is

• A. ${}^{10}{C}_4.\frac{1}{x}$
• B. ${}^{10}{C}_5$
• C. ${}^{10}{C}_5.\frac{1}{x}$
• D. ${}^{10}{C}_6.x$

Answer 1

The correct option is B ${}^{10}{C}_5$

Middle term $=\frac{n}{2}+1$ $=6^{th}$ term
Now consider the following:
$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$...(i)
Where $T$ represents the term.
Here the middle term is the $6^{th}$ term.

Hence $r+1=6$
$\therefore r=5$
substituting in (i), we get
${}^{10}{C}_5{x}^{10-5}\frac{1}{x^5}$
$={}^{10}{C}_5{x}^5{x}^{-5}$
$={}^{10}{C}_5$
Hence, answer is B.