The middle term in the expansion of ${\left(1+x\right)}^{2n}$  is:

Given expansion ${\left(1+x\right)}^{2n}$  is

We have general term in the expansion ${\left(x+a\right)}^n$

$(\therefore T_{r+1}={}^n{\text{C}}_r{x}^{n-r}{\left(a\right)}^r$ be the expansion of ${(x+a)}^n)$

Here, index $=2n$ , which is an even number, therefore, the middle term will be ${\left(n+1\right)}^{th}$  term.

Now, $T_{n+1}={}^{2n}{C}_n{\left(1\right)}^{2n-n}{x}^n$

$T_{n+1}={}^{2n}{C}_n{\left(1\right)}^{2n-n}{x}^n=\frac{2n!}{n!n!}{x}^n$ $\begin{array}{c}({\therefore }^n{C}_r=\frac{n!}{(n-r)!r!})\\ \end{array}$