Solution:

Given:
a * b * c = 15 .........(1)
a² * b² * c² = 83 .......(2)

To find:
a³ * b³ * c³ - 3abc

Let's break down the solution into the following steps:

1. Finding the value of a, b, and c:
From equation (1), we can see that a * b * c = 15. Therefore, at least one of the variables (a, b, or c) must be a factor of 15.

To find the possible combinations of a, b, and c, let's list down all the factors of 15:
Factors of 15: 1, 3, 5, 15

Now, let's try different combinations to satisfy equation (1):

Combination 1: a = 1, b = 3, c = 5
1 * 3 * 5 = 15 (satisfies equation 1)

Combination 2: a = 3, b = 1, c = 5
3 * 1 * 5 = 15 (satisfies equation 1)

Combination 3: a = 5, b = 1, c = 3
5 * 1 * 3 = 15 (satisfies equation 1)

Combination 4: a = 15, b = 1, c = 1
15 * 1 * 1 = 15 (satisfies equation 1)

Therefore, we have four possible combinations of a, b, and c that satisfy equation (1): (1, 3, 5), (3, 1, 5), (5, 1, 3), and (15, 1, 1).

2. Finding the value of a² * b² * c²:
Using equation (2), we can substitute the values of a, b, and c from the four combinations we found in step 1 and calculate the value of a² * b² * c² for each combination:

Combination 1: a = 1, b = 3, c = 5
(1²) * (3²) * (5²) = 1 * 9 * 25 = 225

Combination 2: a = 3, b = 1, c = 5
(3²) * (1²) * (5²) = 9 * 1 * 25 = 225

Combination 3: a = 5, b = 1, c = 3
(5²) * (1²) * (3²) = 25 * 1 * 9 = 225

Combination 4: a = 15, b = 1, c = 1
(15²) * (1²) * (1²) = 225 * 1 * 1 = 225

Therefore, the value of a² * b² * c² is 225 for all four combinations.

3. Finding the value of a³ * b³ * c³ - 3abc:
Using the values of a, b, and c from the four combinations, we can calculate the value of a³ * b³ * c³ - 3abc