The interval in which $x (> 0)$ must lie so that the greatest term in the expansion of ${(1 + x)}^{2n}$n has the greatest coefficient is

The greatest coefficient in the expansion of 

$(1+x{)}^{2n}$ is $2n{C}_n$ We are given ${ }^{2n}{C}_n{x}^n$ is the greatest  term 

$\therefore {}^{2n}{C}_{n-1}{x}^{n-1}{<}^{2n}{C}_n{x}^n$

$\begin{array}{l}{ }^{2n}{C}_{n+1}{x}^{n+1}{<}^{2n}{C}_n{x}^n\\ \Rightarrow \frac{{ }^{2n}{C}_{n-1}}{{ }^{2n}{C}_n}<x<\frac{{ }^{2n}{C}_n}{{ }^{2n}{C}_{n+1}}\\ \Rightarrow \frac{n}{n+1}<x<\frac{n+1}{n}.\end{array}$