If a + b + c + d = 4 and none of a, b, c, d is equal to 1, then find the value of the expression 1/[(1 − a)(1 − b)(1 − c)] + 1/[(1 − b)(1 − c)(1 − d)] + 1/[(1 − c)(1 − d)(1 − a)] + 1/[(1 − d)(1 − a)(1 − b)].

Introduction / Context:
This is an algebraic simplification problem involving four variables and a symmetric sum of rational expressions. The given condition a + b + c + d = 4 is crucial and implies a special relationship when combined with terms of the form 1 − a, 1 − b, and so on. The expression looks complicated but reduces to a simple constant when the symmetry is exploited properly.

Given Data / Assumptions:

• a, b, c, d are real numbers with a + b + c + d = 4.
• None of a, b, c, d equals 1, so denominators like 1 − a are nonzero.
• Expression to evaluate: 1/[(1 − a)(1 − b)(1 − c)] + 1/[(1 − b)(1 − c)(1 − d)] + 1/[(1 − c)(1 − d)(1 − a)] + 1/[(1 − d)(1 − a)(1 − b)].

Concept / Approach:
Introduce new variables A = 1 − a, B = 1 − b, C = 1 − c and D = 1 − d to exploit the symmetry. The given sum then becomes an expression in A, B, C and D. The condition a + b + c + d = 4 can be converted into a condition on A, B, C and D, which turns out to be A + B + C + D = 0. This simplifies the analysis significantly.

Step-by-Step Solution:
Define A = 1 − a, B = 1 − b, C = 1 − c, D = 1 − d. Then a = 1 − A, b = 1 − B, c = 1 − C, d = 1 − D. Use the condition a + b + c + d = 4. Substitute: (1 − A) + (1 − B) + (1 − C) + (1 − D) = 4. This simplifies to 4 − (A + B + C + D) = 4. Therefore A + B + C + D = 0. Rewrite the original expression in terms of A, B, C, D: S = 1/(ABC) + 1/(BCD) + 1/(CDA) + 1/(DAB). Factor out 1/(ABCD): S = (1/(ABCD)) [D + A + B + C]. But A + B + C + D = 0, so the bracket is zero. Hence S = (1/(ABCD)) · 0 = 0.
Verification / Alternative check:
Take a simple example that satisfies a + b + c + d = 4 and avoids 1. For instance, choose a = b = c = d = 1, which is not allowed because denominators vanish. Instead, choose a = b = c = 2 and d = −2; then a + b + c + d = 4. Substitute into the original expression and compute numerically to confirm that the sum is 0.

Why Other Options Are Wrong:
Options like 1 or 4 would require the bracket D + A + B + C to be nonzero and constant, which contradicts the derived condition A + B + C + D = 0. The expression 1 + abcd would make the value depend on the product abcd, but the derived sum is independent of the particular values, as long as the sum a + b + c + d is 4.

Common Pitfalls:
One common mistake is to attempt to combine all four fractions directly over a massive common denominator, which is tedious and error-prone. The smarter approach is to exploit symmetry via the substitution A = 1 − a and the sum condition. Another pitfall is to ignore the constraint a + b + c + d = 4, which is essential to obtaining the simple answer.

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