In a certain arithmetic code, it is given that 9 $ 4 = 17 and 1 $ 5 = 10. Using the same definition of the operation "$", what is the value of 3 $ 6?

Introduction / Context:
This question defines a custom arithmetic operation "$" by giving its values for two pairs of numbers, and then asks you to compute it for a third pair. The goal is to deduce a simple arithmetic rule that maps the inputs to the given outputs. This is common in reasoning tests where you must identify a formula from examples, then apply it to new numbers.

Given Data / Assumptions:

• 9 $ 4 = 17.
• 1 $ 5 = 10.
• The same rule must be used to compute 3 $ 6.
• We assume a straightforward arithmetic relationship involving the two operands.

Concept / Approach:
We look for a simple pattern that uses the two numbers on the left-hand side to produce the result on the right. A natural guess is a linear expression like a + b + constant or a * b plus or minus something. By testing small hypotheses, we can quickly see which formula satisfies both given examples. Once we have the correct rule, we apply it to 3 $ 6.

Step-by-Step Solution:
Step 1: Try a simple addition-based rule: suppose a $ b = a + b + k for some constant k. Step 2: For 9 $ 4 = 17, we have 9 + 4 + k = 17 → 13 + k = 17 → k = 4. Step 3: Check this k with the second example 1 $ 5 = 10: 1 + 5 + 4 = 10, which is correct. Step 4: Therefore, the rule a $ b = a + b + 4 works for both given examples. Step 5: Now compute 3 $ 6 using this rule: 3 + 6 + 4 = 13.

Verification / Alternative check:
We validated the rule for both given examples. An alternative approach would be to test a multiplication-based hypothesis such as a * b - constant, but 9 * 4 = 36 and 1 * 5 = 5 cannot be reduced to 17 and 10 by the same simple constant. Therefore, the addition-plus-4 rule we found is the simplest consistent rule. Because it fits all provided data and there is no contradictory example, we accept it and use it to evaluate 3 $ 6.

Why Other Options Are Wrong:
Given the rule a $ b = a + b + 4, 3 $ 6 must equal 3 + 6 + 4 = 13. Any other value would violate this rule. Options A (11), B (14), D (16) and E (12) do not match the result of the correctly inferred operation. Since the operation definition must be consistent across all uses, those values cannot be correct.

Common Pitfalls:
Candidates sometimes jump to more complicated formulas unnecessarily (involving products or squares) and get lost in algebra. Another common mistake is to accept a formula that works for the first example but not check it against the second. Always test your hypothesis with all given pairs to ensure consistency before applying it to a new pair.

Final Answer:
Therefore, according to the pattern a $ b = a + b + 4, the value of 3 $ 6 is 13.