Three distinct numbers are selected randomly from the set \ 1,2,3,...

$1 \leq \mathrm{a} \lt \mathrm{ar} \lt \mathrm{ar}^2 \leq 40$
(If $r \in N$ )
If $r=2$
$1 \leq a \lt 2 a \lt 4 a \leq 40$
$a \in\{1, \ldots \ldots ., 10\}$ ________ (10 GP)
If $r=3$
$1 \leq a \lt 3 a \lt 9 a \leq 40$
$\mathrm{a} \in\{1,2,3,4\}$ ________ ________ (4 GP)
If $\mathrm{r}=4$
$1 \leq a \lt 4 a \lt 16 a \leq 40$
$a \in\{1,2\}$ ________ ________ (2 GP)
If $r=5$
$1 \leq a \lt 5 a \lt 25 a \leq 40$
$a \in\{1\}$ ________ ________ (1 GP)
If $r=6$
$1 \leq a \lt 6 a \lt 36 a \leq 40$
$\mathrm{a} \in\{1\}$ ________ ________ (1 GP)
\left(\mathrm{P}=\frac{18}{9880}=\frac{9}{4940}\right)$as per NTA for$\mathrm{r} \in \mathrm{N}$<br/>$\mathrm{m}+\mathrm{n}=4949$<br/>If$\underline{r} \notin \mathrm{~N}$(also possible)<br/>$\mathrm{r}=\frac{3}{2}$<br/>$\operatorname{ar}^2=\frac{9 \mathrm{a}}{4} ; \mathrm{a}=4 \mathrm{k}<br/>\left.$\begin{array}{l}(4,6,9) \\ (8,12,18) \\ (12,18,27) \\ (16,24,36)\end{array}$\right\} 4 \mathrm{GP}
$$\begin{aligned} & \begin{array}{l}\mathrm{r}=\frac{5}{2} \quad \mathrm{ar}^2=\frac{25 \mathrm{a}}{4} ; \mathrm{a}=4 \mathrm{k} \\ \\ \\ (4,10,25) \ldots \ldots . .(1) \mathrm{GP} \\ \mathrm{r}=\frac{4}{3} \quad \mathrm{ar}^2=\frac{16 \mathrm{a}}{9} \rightarrow \mathrm{a}=9 \mathrm{k} \\ \\ \\ (9,12,16),(18,24,32) \ldots \ldots .(2) \mathrm{GP} \\ \mathrm{r}=\frac{5}{3} \quad \mathrm{ar}^2=\frac{25 \mathrm{a}}{9} ; \mathrm{a}=9 \mathrm{k} \\ \\ \quad(9,15,25) \ldots \ldots \ldots .(1) \mathrm{GP} \\ \mathrm{r}=\frac{5}{4} \quad \mathrm{ar}^2=\frac{25 \mathrm{a}}{16} ; \mathrm{a}=16 \mathrm{k}\end{array} \\ & \quad \begin{array}{l}(16,20,25) \ldots \ldots \ldots . .(1) \mathrm{GP} \\ \mathrm{r}=\frac{6}{5} \quad \mathrm{ar}^2=\frac{36 \mathrm{a}}{25} ; \mathrm{a}=25 \mathrm{k}\end{array} \\ & \begin{array}{l}(25,30,36) \ldots \ldots \ldots . .(1) \mathrm{GP} \\ \mathrm{Total}=18+10=28 \\ \mathrm{P}=\frac{28}{40} \mathrm{C}_3=\frac{28}{9880}=\frac{7}{2470} \\ \mathrm{~m}+\mathrm{n}=2477\end{array}\end{aligned}$$