Simplify the identity: [ (m − n)^3 + (n − r)^3 + (r − m)^3 ] ÷ [ 6 (m − n)(n − r)(r − m) ]. Evaluate the constant value independent of m, n, r.

Introduction / Context:
This expression is a textbook application of the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). A clever choice of a, b, c makes the sum vanish, drastically simplifying the fraction to a constant.

Given Data / Assumptions:

• a = m − n, b = n − r, c = r − m.
• We must compute [a^3 + b^3 + c^3] / [6abc] with these substitutions.

Concept / Approach:
Note that a + b + c = (m − n) + (n − r) + (r − m) = 0. Plugging into the cubic identity yields a^3 + b^3 + c^3 − 3abc = 0 * ( … ) = 0, hence a^3 + b^3 + c^3 = 3abc. Substitute back into the ratio and simplify to a constant independent of m, n, r.

Step-by-Step Solution:
Compute a + b + c = 0.From identity: a^3 + b^3 + c^3 − 3abc = (a + b + c)(… ) = 0 ⇒ a^3 + b^3 + c^3 = 3abc.Therefore the required value = (3abc) / (6abc) = 1/2.

Verification / Alternative check:
Pick simple numbers, e.g., m = 2, n = 1, r = 0 ⇒ a = 1, b = 1, c = −2. Then numerator 1 + 1 + (−8) = −6; denominator 6*(1*1*(−2)) = −12; ratio = (−6)/(−12) = 1/2, confirming the identity.

Why Other Options Are Wrong:

• 1/3, 1/6, 1/5, 2/3: These come from forgetting the factor 3 in a^3 + b^3 + c^3 = 3abc or miscomputing the denominator.

Common Pitfalls:
Applying the identity but not checking that a + b + c = 0; sign mistakes in (r − m).

Final Answer:
1/2