Solution:

To find the probability that the selected numbers form a geometric progression (G.P), we first need to determine the total number of possible outcomes.

Total number of ways to choose 3 numbers from the first 20 natural numbers =
^20C₃ = (20 * 19 * 18) / (3 * 2 * 1) = 1140

Now, we need to find the number of favorable outcomes, i.e., the number of ways in which we can choose 3 numbers that form a G.P.

A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a constant ratio. Let's assume the three chosen numbers are a, ar, and ar^2.

To form a G.P, we need to consider all possible values of 'a', 'r', and 'r^2' within the given range of the first 20 natural numbers.

- Values of 'a':
- If 'a' is 1, then 'r' can be any number from 2 to 10 (as r^2 should be within 20).
- If 'a' is 2, then 'r' can be any number from 2 to 5 (as r^2 should be within 20).
- If 'a' is 3, then 'r' can be any number from 2 to 3 (as r^2 should be within 20).
- If 'a' is 4, then 'r' can be 2 (as r^2 should be within 20).
- If 'a' is 5, then 'r' can be 2 (as r^2 should be within 20).
- If 'a' is 6, then 'r' can be 2 (as r^2 should be within 20).
- If 'a' is 7, then 'r' can be 2 (as r^2 should be within 20).
- If 'a' is 8, then 'r' can be 2 (as r^2 should be within 20).
- If 'a' is 9, then 'r' can be 2 (as r^2 should be within 20).
- If 'a' is 10, then 'r' can be 2 (as r^2 should be within 20).

- Values of 'r':
- If 'r' is 2, then 'a' can be any number from 1 to 10 (as ar^2 should be within 20).
- If 'r' is 3, then 'a' can be any number from 1 to 3 (as ar^2 should be within 20).
- If 'r' is 4, then 'a' can be 1 (as ar^2 should be within 20).
- If 'r' is 5, then 'a' can be 1 (as ar^2 should be within 20).
- If 'r' is 6, then 'a' can be 1 (as ar^2 should be within 20).
- If 'r' is 7, then 'a' can be 1 (as ar^2 should be within 20).
- If 'r