To find the number of distinct positive rational numbers p/q smaller than 1, where p, q < 20,="" we="" can="" use="" a="" two-step="" />

Step 1: Generate all possible pairs of positive integers (p, q) where p < q="" />< />

Step 2: Reduce each pair (p, q) to its simplest form by dividing both p and q by their greatest common divisor (GCD).

Step 1: Generating all possible pairs (p, q):
Starting with p = 1, we can generate all possible pairs (p, q) where p < q="" />< 20="" by="" incrementing="" p="" and="" q="" in="" a="" nested="" />

For p = 1:
- q = 2, 3, 4, ..., 19
For p = 2:
- q = 3, 4, 5, ..., 19
For p = 3:
- q = 4, 5, 6, ..., 19
...
For p = 18:
- q = 19

The total number of pairs generated can be calculated using the formula (20 - p - 1) + (20 - p - 2) + ... + 1, which simplifies to (20 - p - 1) * (20 - p) / 2.

Step 2: Reducing each pair (p, q) to its simplest form:
For each pair (p, q), we need to calculate the GCD of p and q using the Euclidean algorithm, and then divide both p and q by their GCD.

The number of distinct positive rational numbers p/q smaller than 1, where p, q < 20,="" will="" be="" the="" count="" of="" distinct="" reduced="" pairs="" (p,="" q)="" obtained="" in="" step="" />

Here is a Python code to implement this approach:

```python
import math

count = 0

for p in range(1, 20):
for q in range(p + 1, 20):
# Step 1: Generate all possible pairs (p, q)

# Step 2: Reduce each pair (p, q) to its simplest form
gcd = math.gcd(p, q)
reduced_p = p // gcd
reduced_q = q // gcd

# Increment count if the reduced pair is distinct
if (reduced_p, reduced_q) not in distinct_pairs:
count += 1

print(count)
```

The output of this code will be the number of distinct positive rational numbers p/q smaller than 1, where p, q < 20.="" />