The middle term in the expansion of ${\left(1-\frac{1}{x}\right)}^n(1-x{)}^n,$ is 

We have,

${\left(1-\frac{1}{x}\right)}^n(1-x{)}^n=(1-x{)}^{2n}\frac{(-1{)}^n}{x^n}=\frac{(-1{)}^n(1-x{)}^{2n}}{x^n}$

$\therefore$ Middle term in ${\left(1-\frac{1}{x}\right)}^n(1-x{)}^n$

$\begin{array}{l}=\frac{(-1{)}^n}{x^n}-\text{ Middle term in }(1-x{)}^{2n}\\ =\frac{(-1{)}^n}{x^n}\times (n+1{)}^{th}\text{ term in }(1-x{)}^{2n}\\ =\frac{(-1{)}^n}{x^n}{\times }^{2n}{C}_n(-x{)}^n{=}^{2n}{C}_n.\end{array}$