Let x, y and z be real numbers such that x + y + z = 9. Evaluate the expression (x - 4)^3 + (y - 2)^3 + (z - 3)^3 - 3(x - 4)(y - 2)(z - 3).

Introduction / Context:
This problem is designed to test your understanding of the classic cubic identity u^3 + v^3 + w^3 - 3uvw. It appears frequently in algebra sections of aptitude exams, usually in a disguised form with shifted variables, as in x - 4, y - 2 and z - 3 here.

Given Data / Assumptions:
- x, y and z are real numbers.
- x + y + z = 9.
- We must evaluate (x - 4)^3 + (y - 2)^3 + (z - 3)^3 - 3(x - 4)(y - 2)(z - 3).

Concept / Approach:
The important identity is u^3 + v^3 + w^3 - 3uvw = (u + v + w)(u^2 + v^2 + w^2 - uv - vw - wu). A special case occurs when u + v + w = 0. In that case the entire expression collapses to zero, no matter what the individual values of u, v and w are.

Step-by-Step Solution:
Define u = (x - 4), v = (y - 2) and w = (z - 3). Then the given expression is u^3 + v^3 + w^3 - 3uvw. First find u + v + w: u + v + w = (x - 4) + (y - 2) + (z - 3). This simplifies to x + y + z - (4 + 2 + 3) = (x + y + z) - 9. Using x + y + z = 9, we get u + v + w = 9 - 9 = 0. By the identity, if u + v + w = 0, then u^3 + v^3 + w^3 - 3uvw = 0. Therefore, the value of (x - 4)^3 + (y - 2)^3 + (z - 3)^3 - 3(x - 4)(y - 2)(z - 3) is 0.

Verification / Alternative check:
You can verify with a simple numeric example consistent with x + y + z = 9, for instance x = 4, y = 2, z = 3. Then u = 0, v = 0, w = 0, and the expression clearly becomes 0^3 + 0^3 + 0^3 - 3 * 0 * 0 * 0 = 0, matching the identity based result.

Why Other Options Are Wrong:
Values like 6, 9, 1 or 3 might result if you substitute random numbers that do not respect the condition on x + y + z, or if you do not recognise the special case u + v + w = 0 and attempt partial expansion, losing terms or making arithmetic errors.

Common Pitfalls:
A common mistake is to expand each cube fully, which is very time consuming and error prone. Another pitfall is to forget to group the expression as u^3 + v^3 + w^3 - 3uvw in a consistent way, so the identity cannot be applied cleanly. Quickly checking the sum u + v + w before expanding saves a lot of effort.

Final Answer:
The value of the given expression is 0.