If the coefficients of xⁿ in the expansion of ${\left(1+x\right)}^{2n}$ and ${\left(1+x\right)}^{2n-1}$ are A and B respectively then $A=\lambda B$, then $\lambda =$

A = coefficient of $x^n\text{ in }(1+x{)}^{2n}{=}^{2n}{C}_n$ 
B = coefficient in  $x^n\text{ in }(1+x{)}^{2n-1}{=}^{2n-1}{C}_n$
$\therefore \frac{\mathrm{A}}{\mathrm{B}}=\frac{{ }^{2\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}}{{ }^{2\mathrm{n}-1}{\mathrm{C}}_{\mathrm{n}}}=\frac{{\displaystyle \frac{2n}{n}}\left({ }^{2\mathrm{n}-1}{\mathrm{C}}_{\mathrm{n}-1}\right)}{{ }^{2\mathrm{n}-1}{\mathrm{C}}_{\mathrm{n}}}\Rightarrow \frac{\mathrm{A}}{\mathrm{B}}=2\therefore \mathrm{A}=2\mathrm{B}$