Coefficient of the term independent of x in the expan-sion of $(1+x{)}^{2n}{\left(\frac{x}{1-x}\right)}^{-2n}$ is :

Coefficient of the term independent of x in the expan-sion of

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

or $\frac{(1+x{)}^{2n}(1-x{)}^{2n}}{x^{2n}}$ 

= coefficient of $x^{2n}\text{ in }{\left(1-x^2\right)}^{2n}$

= coefficient of $t^n\text{ in }(1-t{)}^{2n}$

${=}^{2n}{C}_n(-1{)}^n$