Ratio of coefficients of ${\mathrm{x}}^{18}{\mathrm{y}}^2$ and ${\mathrm{x}}^{17}{\mathrm{y}}^3$ in the expansion of ${\left(\frac{2\mathrm{x}}{3}+\frac{3\mathrm{y}}{2}\right)}^{20}$ is $\frac{\mathrm{m}}{\mathrm{n}}$(where m and n are natural numbers and in simplest form), then $\mathrm{m}+\mathrm{n}$ is

General term ${\mathrm{T}}_{\mathrm{r}+1}{=}^{20}{\mathrm{C}}_{\mathrm{r}}{\left(\frac{{\displaystyle 2\mathrm{x}}}{{\displaystyle 3}}\right)}^{20-\mathrm{r}}{\left(\frac{{\displaystyle 3\mathrm{y}}}{{\displaystyle 2}}\right)}^{\mathrm{r}}$
$\Rightarrow {\mathrm{T}}_{\mathrm{r}+1}{=}^{20}{\mathrm{C}}_{\mathrm{r}}{\left(\frac{{\displaystyle 2}}{{\displaystyle 3}}\right)}^{20-2\mathrm{r}}{\mathrm{x}}^{20-\mathrm{r}}{\mathrm{y}}^{\mathrm{r}}...............\left(1\right)$
Put $\Rightarrow \mathrm{r}=2\hspace{0.17em}\hspace{0.17em}\mathrm{in}\hspace{0.17em}\hspace{0.17em}\left(1\right)$
${\mathrm{T}}_3{=}^{20}{\mathrm{C}}_2{\left(\frac{{\displaystyle 2}}{{\displaystyle 3}}\right)}^{16}.{\mathrm{x}}^{18}.{\mathrm{y}}^2$
$\therefore$coefficient of ${\mathrm{x}}^{18}{\mathrm{y}}^{12}$ is ${=}^{20}{\mathrm{C}}_2{\left(\frac{{\displaystyle 2}}{{\displaystyle 3}}\right)}^{16}$
Put $\mathrm{r}=3\mathrm{in}\left(1\right)$
${\mathrm{T}}_4{=}^{20}{\mathrm{C}}_3{\left(\frac{{\displaystyle 2}}{{\displaystyle 3}}\right)}^{14}{\mathrm{x}}^{17}{\mathrm{y}}^3$
$\therefore$coefficient of ${\mathrm{x}}^{17}{\mathrm{y}}^3 \mathrm{is}{ }^{20}{\mathrm{C}}_3{\left(\frac{2}{3}\right)}^{14}$
Reqd. ratio $=\frac{{ }^{20}{\mathrm{C}}_2{\left(\frac{2}{3}\right)}^{16}}{{ }^{20}{\mathrm{C}}_3{\left(\frac{2}{3}\right)}^{14}}=\frac{1}{6}\times {\left(\frac{{\displaystyle 2}}{{\displaystyle 3}}\right)}^2$
$=\frac{2}{27}$