1. Total number of ways to choose 3 distinct numbers from $\{1,2,\dots,40\}$:

Since the numbers are selected and then checked whether they are in an increasing G.P., each valid set of 3 distinct numbers corresponds to exactly one increasing triple.

2. Form of a 3-term increasing G.P.

Let the three numbers be

with $a, ar, ar^2 \in \{1,2,\dots,40\}$ and $r>1$.

Because all three terms are integers, write the common ratio in lowest terms as

Then the terms become

For these to all be integers, $a$ must be divisible by $q^2$. So let

Then the three terms are

Thus every increasing integer G.P. has the form

where $p>q$, $\gcd(p,q)=1$, and

So for each coprime pair $(p,q)$ with $p>q$, the number of valid choices of $k$ is

Also, for a fixed triple, the reduced ratio $p/q$ is unique, so there is no overcounting.

3. Find all possible values of $p$

Since $kp^2\le 40$ and $k\ge 1$, we need

So we check $p=2,3,4,5,6$.

4. Count valid triples for each $p$

Case $p=2$

Possible $q<2$ with $\gcd(2,q)=1$:

• $q=1$

Number of $k$:

Contribution: $1\times 10=10$

Case $p=3$

Possible $q<3$ with $\gcd(3,q)=1$:

• $q=1,2$

Number of $k$:

Contribution: $2\times 4=8$

Case $p=4$

Possible $q<4$ with $\gcd(4,q)=1$:

• $q=1,3$

Number of $k$:

Contribution: $2\times 2=4$

Case $p=5$

Possible $q<5$ with $\gcd(5,q)=1$:

• $q=1,2,3,4$

Number of $k$:

Contribution: $4\times 1=4$

Case $p=6$

Possible $q<6$ with $\gcd(6,q)=1$:

• $q=1,5$

Number of $k$:

Contribution: $2\times 1=2$

5. Total favorable selections

So the required probability is

Here,

Therefore,

6. Comparison with stored answer

Stored correct answer = $2477$

Our derived answer is also $2477$. Hence they agree.