In the following question, each option shows a pair of numbers separated by a comma. In three of the pairs the second number is one less than the cube of the first number. Select the pair that does not follow this rule (odd pair).

Introduction / Context:
This numerical reasoning question involves a relationship between two numbers in each pair based on cubes. Odd one out questions of this type are used in aptitude exams to test whether candidates can quickly identify arithmetic patterns like squares and cubes. Recognising these patterns helps in many areas of mathematics, including sequences, algebra, and problem solving.

Given Data / Assumptions:
- The options are the pairs 5,124; 7,342; 3,26; 2,15; and 4,63.
- The first number in each pair is a small positive integer.
- The second number is related to the cube of the first number in some way.
- Exactly one pair does not follow the same rule that the other pairs follow.

Concept / Approach:
The pattern described is that in three of the pairs, the second number equals the cube of the first number minus 1. That is, if the first number is n, the second number should be n^3 - 1. To apply this approach, we compute the cube of the first number in each pair and then subtract 1, and compare the result with the given second number. The pair in which this relation does not hold will be the odd one out.

Step-by-Step Solution:
Step 1: For the pair 5,124, compute the cube of 5. We get 5^3 = 5 * 5 * 5 = 125. Subtract 1 to get 125 - 1 = 124, which matches the second number.Step 2: For 7,342, compute the cube of 7. We get 7^3 = 343. Subtract 1 to get 343 - 1 = 342, which again matches the second number.Step 3: For 3,26, compute the cube of 3. We get 3^3 = 27. Subtract 1 to get 27 - 1 = 26, which matches the second number.Step 4: For 4,63, compute the cube of 4. We get 4^3 = 64. Subtract 1 to get 64 - 1 = 63, which matches the second number.Step 5: For 2,15, compute the cube of 2. We get 2^3 = 8. Subtract 1 to get 8 - 1 = 7, which does not match the second number 15.Step 6: Since four pairs correctly follow the rule second number = first number cubed minus 1, and only the pair 2,15 fails to match this rule, 2,15 is the odd one out.

Verification / Alternative Check:
We can verify this pattern by writing the relationship explicitly for each valid pair. For 5,124 we have 124 = 5^3 - 1. For 7,342 we have 342 = 7^3 - 1. For 3,26 we have 26 = 3^3 - 1. For 4,63 we have 63 = 4^3 - 1. For 2,15 the relation would require 15 to be equal to 2^3 - 1, but 2^3 - 1 = 7, not 15. Therefore this pair does not satisfy the same mathematical rule and is correctly identified as the odd one out.

Why Other Options Are Wrong:
5,124 is not wrong because it fits the pattern exactly, with 124 equal to 5^3 - 1. 7,342 also fits perfectly since 342 equals 7^3 - 1. 3,26 matches the rule with 26 equal to 3^3 - 1. 4,63 again respects the relation because 63 equals 4^3 - 1. All of these share the same cube minus one structure and so belong to the main group of consistent pairs.

Common Pitfalls:
A common mistake is to assume that the relation is second number equals first number squared or some other operation, and then to give up when the numbers do not match. Another pitfall is doing cube calculations incorrectly under time pressure. To avoid errors, always compute n^3 carefully, then subtract 1, and compare the result with the given second number. Understanding the cube relation used in this question improves overall skill with powers and patterns in aptitude tests.

Final Answer:
2,15