In aptitude (pattern based simplification), a custom operation * is defined so that 9 * 6 = 45, 7 * 4 = 33, and 6 * 4 = 20. Using the same hidden rule, find the value of 5 * 3.

Introduction / Context:
This question is a pattern based or rule based operation problem. The operation * is not standard multiplication; instead, it is defined by a hidden pattern which we must discover using the given examples. Such problems are common in reasoning and aptitude tests, designed to check logical thinking and pattern recognition skills rather than pure arithmetic knowledge.

Given Data / Assumptions:
- 9 * 6 = 45.
- 7 * 4 = 33.
- 6 * 4 = 20.
- The same rule must be used to find 5 * 3.
- The operation * is assumed to be symmetric in the examples (but we must infer the rule from the data provided).

Concept / Approach:
The key idea is to express the result of each operation in terms of the two input numbers a and b and then look for a consistent formula. A good approach is to try combinations such as a * b = (a + b)(a - b), or a * b = a^2 - b^2, or a * b = a + b + ab, and so on. We test each hypothesis against the three given examples. Once we find a rule that works for all given cases, we apply that rule to compute 5 * 3.

Step-by-Step Solution:
Step 1: Consider the possibility that a * b relates to (a + b)(a - b).Step 2: For a = 9, b = 6: (9 + 6)(9 - 6) = 15 * 3 = 45, which matches 9 * 6 = 45.Step 3: For a = 7, b = 4: (7 + 4)(7 - 4) = 11 * 3 = 33, which matches 7 * 4 = 33.Step 4: For a = 6, b = 4: (6 + 4)(6 - 4) = 10 * 2 = 20, which matches 6 * 4 = 20.Step 5: Since all three examples fit the pattern, we conclude that a * b = (a + b)(a - b).Step 6: Apply the rule to 5 * 3: (5 + 3)(5 - 3) = 8 * 2 = 16.Step 7: Therefore, 5 * 3 = 16 under this custom operation.

Verification / Alternative check:
We could also recognise that (a + b)(a - b) is equal to a^2 - b^2. Checking with the given examples: for 9 and 6, a^2 - b^2 = 81 - 36 = 45; for 7 and 4, 49 - 16 = 33; for 6 and 4, 36 - 16 = 20. All are consistent. For 5 and 3, a^2 - b^2 = 25 - 9 = 16, matching the result from the previous method. This double confirmation reinforces that the correct rule has been found.

Why Other Options Are Wrong:
- 24 and 12: These might come from naive multiplication or addition patterns such as a * b = a * b + something, but they do not fit the rule that matches all three given examples.
- 13 and 10: These may be guessed values based on partial observation or incorrect pattern discovery, but they do not arise from (a + b)(a - b) or a^2 - b^2 for 5 and 3.

Common Pitfalls:
One common mistake is to assume the operation is simply a * b = a * b (ordinary multiplication) and then try to fit corrections to get the given results, which usually becomes inconsistent. Another pitfall is picking a pattern that works for only one or two examples without checking all of them. In pattern based questions, it is essential to verify that the discovered rule fits every given instance before applying it to a new pair of numbers.

Final Answer:
Based on the discovered rule a * b = (a + b)(a - b), the value of 5 * 3 is 16.