Three distinct numbers are selected randomly from the set 1,2,3, …, 40. If the probability, that the selected numbers are in an increasing G.P. is m…

$$1 \leq \mathrm{a} \lt \mathrm{ar} \lt \mathrm{ar}^2 \leq 40$$1≤a<ar<ar2≤40

(If $r \in N$r∈N )

If $r=2$r=2

$$1 \leq a \lt 2 a \lt 4 a \leq 40$$1≤a<2a<4a≤40

$a \in\{1, \ldots \ldots ., 10\}$a∈{1,…….,10} ________ (10 GP)

If $r=3$r=3

$$1 \leq a \lt 3 a \lt 9 a \leq 40$$1≤a<3a<9a≤40

$\mathrm{a} \in\{1,2,3,4\}$a∈{1,2,3,4} ________ ________ (4 GP)

If $\mathrm{r}=4$r=4

$$1 \leq a \lt 4 a \lt 16 a \leq 40$$1≤a<4a<16a≤40

$a \in\{1,2\}$a∈{1,2} ________ ________ (2 GP)

If $r=5$r=5

$$1 \leq a \lt 5 a \lt 25 a \leq 40$$1≤a<5a<25a≤40

$a \in\{1\}$a∈{1} ________ ________ (1 GP)

If $r=6$r=6

$$1 \leq a \lt 6 a \lt 36 a \leq 40$$1≤a<6a<36a≤40

$\mathrm{a} \in\{1\}$a∈{1} ________ ________ (1 GP)

$\left(\mathrm{P}=\frac{18}{9880}=\frac{9}{4940}\right)$(P=988018​=49409​) as per NTA for $\mathrm{r} \in \mathrm{N}$r∈N

$$\mathrm{m}+\mathrm{n}=4949$$m+n=4949

If $\underline{r} \notin \mathrm{~N}$r​∈/ N (also possible)

$$\mathrm{r}=\frac{3}{2}$$r=23​

$$\operatorname{ar}^2=\frac{9 \mathrm{a}}{4} ; \mathrm{a}=4 \mathrm{k}$$ar2=49a​;a=4k

$$\left.\begin{array}{l}(4,6,9) \\ (8,12,18) \\ (12,18,27) \\ (16,24,36)\end{array}\right\} 4 \mathrm{GP}$$(4,6,9)(8,12,18)(12,18,27)(16,24,36)​⎭⎬⎫​4GP

$$\begin{aligned} & \begin{array}{l}\mathrm{r}=\frac{5}{2} \quad \mathrm{ar}^2=\frac{25 \mathrm{a}}{4} ; \mathrm{a}=4 \mathrm{k} \\ \\ \\ (4,10,25) \ldots \ldots . .(1) \mathrm{GP} \\ \mathrm{r}=\frac{4}{3} \quad \mathrm{ar}^2=\frac{16 \mathrm{a}}{9} \rightarrow \mathrm{a}=9 \mathrm{k} \\ \\ \\ (9,12,16),(18,24,32) \ldots \ldots .(2) \mathrm{GP} \\ \mathrm{r}=\frac{5}{3} \quad \mathrm{ar}^2=\frac{25 \mathrm{a}}{9} ; \mathrm{a}=9 \mathrm{k} \\ \\ \quad(9,15,25) \ldots \ldots \ldots .(1) \mathrm{GP} \\ \mathrm{r}=\frac{5}{4} \quad \mathrm{ar}^2=\frac{25 \mathrm{a}}{16} ; \mathrm{a}=16 \mathrm{k}\end{array} \\ & \quad \begin{array}{l}(16,20,25) \ldots \ldots \ldots . .(1) \mathrm{GP} \\ \mathrm{r}=\frac{6}{5} \quad \mathrm{ar}^2=\frac{36 \mathrm{a}}{25} ; \mathrm{a}=25 \mathrm{k}\end{array} \\ & \begin{array}{l}(25,30,36) \ldots \ldots \ldots . .(1) \mathrm{GP} \\ \mathrm{Total}=18+10=28 \\ \mathrm{P}=\frac{28}{40} \mathrm{C}_3=\frac{28}{9880}=\frac{7}{2470} \\ \mathrm{~m}+\mathrm{n}=2477\end{array}\end{aligned}$$​r=25​ar2=425a​;a=4k(4,10,25)……..(1)GPr=34​ar2=916a​→a=9k(9,12,16),(18,24,32)…….(2)GPr=35​ar2=925a​;a=9k(9,15,25)……….(1)GPr=45​ar2=1625a​;a=16k​(16,20,25)………..(1)GPr=56​ar2=2536a​;a=25k​(25,30,36)………..(1)GPTotal=18+10=28P=4028​C3​=988028​=24707​ m+n=2477​​