Classification – Odd one out: Identify the single number that does not follow the same formation rule as the others: 6, 24, 64, 120.

Introduction / Context:
In odd-one-out classification questions, several items share a common generation rule or mathematical property, while exactly one option violates that rule. Your task is to spot the dissimilar element by testing simple, high-signal properties first (factorials, perfect powers, prime/composite patterns), then moving to secondary patterns if needed.

Given Data / Assumptions:

• Numbers given: 6, 24, 64, 120.
• Assume standard integer properties: factorials n!, perfect squares a^2, perfect cubes b^3, and so on.

Concept / Approach:
Check for recognizable families. Factorials around this range: 3! = 6, 4! = 24, 5! = 120. Also check perfect powers: 64 = 8^2 and also 4^3 = 64. A strong cue is that 6, 24, and 120 are consecutive factorials, a very specific family; anything not in that family is likely the outlier.

Step-by-Step Solution:
Compute 3! = 3 * 2 * 1 = 6 (present in the list).Compute 4! = 4 * 3 * 2 * 1 = 24 (present in the list).Compute 5! = 5 * 4 * 3 * 2 * 1 = 120 (present in the list).Test 64 for factorial membership: there is no integer n with n! = 64.Note: 64 is 2^6 and 4^3, i.e., a perfect power, but not a factorial.

Verification / Alternative check:
Attempt to represent each as a factorial: only 6, 24, 120 succeed. Cross-check by prime-factorizing 64 = 2^6; factorials have a mixed prime structure (e.g., 5! = 2^3 * 3 * 5), not a single prime power. This corroborates that 64 is structurally different.

Why Other Options Are Wrong:

• 6: equals 3!, fits the factorial pattern.
• 24: equals 4!, fits the factorial pattern.
• 120: equals 5!, fits the factorial pattern.

Common Pitfalls:
Seeing that 64 is both a perfect square and cube may distract from the dominant pattern. In classification, prefer the strongest shared property across the majority (here, factorials) over interesting but solitary traits of the remaining item.

Final Answer:
64