Given that a + b + c = 27, evaluate the following expression: (a − 7)^3 + (b − 9)^3 + (c − 11)^3 − 3(a − 7)(b − 9)(c − 11) Choose the correct value.

Introduction / Context: This question is about recognizing a famous algebraic identity: p^3 + q^3 + r^3 − 3pqr, especially when p + q + r = 0. If you spot the pattern, the expression collapses immediately.

Given Data / Assumptions:

• a + b + c = 27 • Expression: (a − 7)^3 + (b − 9)^3 + (c − 11)^3 − 3(a − 7)(b − 9)(c − 11)

Concept / Approach: Let: p = a − 7, q = b − 9, r = c − 11. Then the expression becomes: p^3 + q^3 + r^3 − 3pqr. Also: p + q + r = (a + b + c) − (7 + 9 + 11) = 27 − 27 = 0. Identity: If p + q + r = 0, then p^3 + q^3 + r^3 − 3pqr = 0.

Step-by-Step Solution: 1) Define p = a − 7, q = b − 9, r = c − 11 2) Compute sum: p + q + r = (a + b + c) − 27 = 27 − 27 = 0 3) Recognize identity form: p^3 + q^3 + r^3 − 3pqr 4) Apply identity for p + q + r = 0: p^3 + q^3 + r^3 − 3pqr = 0

Verification / Alternative check: Pick a simple example satisfying a + b + c = 27, such as a = 7, b = 9, c = 11. Then p = q = r = 0 and the expression is clearly 0. This confirms the result matches the identity behavior.

Why Other Options Are Wrong: • 3, 9, 27, 81: these are typical cube-related distractors, but the identity forces the result to be exactly 0.

Common Pitfalls: • Expanding cubes directly (unnecessary and error-prone). • Not noticing that (7 + 9 + 11) equals 27, making p + q + r = 0.

Final Answer: 0