Simplify the algebraic expression 1/[(p − n)(n − q)] + 1/[(n − q)(q − p)] + 1/[(q − p)(p − n)] and find its value in terms of p, q and n.

Introduction / Context:
This question tests simplification of a symmetric rational expression involving three variables p, q and n. Such expressions often appear complicated but simplify nicely when denominators are examined carefully. The structure shows products of differences of the variables, and the sum has a cyclic pattern, which suggests that many terms may cancel out when a common denominator is used.

Given Data / Assumptions:

• Expression: 1/[(p − n)(n − q)] + 1/[(n − q)(q − p)] + 1/[(q − p)(p − n)].

• p, q and n are distinct real numbers so that none of the denominators is zero.

• We must simplify the sum and express it as a single number or simple expression.

Concept / Approach:
The three terms share related factors in the denominators. The natural approach is to bring them over a common denominator, which is the product (p − n)(n − q)(q − p), and then combine numerators. Because of symmetry and alternating signs, there is a strong chance that the combined numerator simplifies to zero. Carefully accounting for signs of (q − p) versus (p − q) and similar pairs is crucial to avoid errors.

Step-by-Step Solution:
Step 1: Identify the common denominator as D = (p − n)(n − q)(q − p). Step 2: Rewrite the first term with denominator D. Multiply numerator and denominator by (q − p), giving (q − p) / D. Step 3: Rewrite the second term. Its denominator is (n − q)(q − p), so multiply numerator and denominator by (p − n), giving (p − n) / D. Step 4: Rewrite the third term. Its denominator is (q − p)(p − n), so multiply numerator and denominator by (n − q), giving (n − q) / D. Step 5: Combine the three numerators: (q − p) + (p − n) + (n − q). Step 6: Simplify the combined numerator: q − p + p − n + n − q = 0. Step 7: Therefore the entire expression becomes 0 / D = 0 (provided D ≠ 0).

Verification / Alternative check:
As a numerical check, choose specific values such as p = 1, q = 2 and n = 3. Then compute each term. We have (p − n)(n − q) = (1 − 3)(3 − 2) = (−2)(1) = −2, so the first term is −1/2. Next, (n − q)(q − p) = (3 − 2)(2 − 1) = (1)(1) = 1, so the second term is 1. Finally, (q − p)(p − n) = (2 − 1)(1 − 3) = (1)(−2) = −2, so the third term is −1/2. The sum is −1/2 + 1 − 1/2 = 0. This matches the algebraic result and confirms the simplification.

Why Other Options Are Wrong:
Option 1 would require the combined numerator to equal the common denominator, which does not happen. Options p + q + n and pq + qn + np would be non zero in general, contradicting both the algebraic simplification and numerical checks. The expression 2np + q is unrelated to the structure of the original sum. Only the value 0 is consistent for all allowed values of p, q and n that keep denominators non zero.

Common Pitfalls:
The most common errors are sign mistakes when handling factors such as (q − p) and (p − q), which are negatives of each other. Some learners also forget to multiply the numerators properly while forming the common denominator. Another pitfall is trying to plug in arbitrary values without considering whether denominators become zero. Following a systematic common denominator approach and simplifying the numerator step by step avoids these issues.

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