In a certain numerical code, the symbol $ represents a special operation between two numbers. It is given that 75 $ 26 = 4 and 69 $ 53 = 7. Using the same rule, what is the value of 82 $ 46 ?

Introduction / Context:
This is a coded operation question where the symbol $ stands for a specific rule rather than ordinary arithmetic. We are given two examples of how this operation behaves on pairs of two digit numbers, and then we must determine the result for a new pair. The challenge is to recognise that the operation is based on the digits of the numbers rather than their whole values, which is a common trick in reasoning questions of this type.

Given Data / Assumptions:

• 75 $ 26 = 4.
• 69 $ 53 = 7.
• The same definition of $ applies to all pairs.
• We must evaluate 82 $ 46.
• Numbers are two digit, so digit based operations like digit sums are natural candidates.

Concept / Approach:
Because the outputs 4 and 7 are much smaller than the inputs, it is likely that the operation uses digit sums rather than direct multiplication or subtraction of the full numbers. A simple and effective hypothesis is that we take the sum of the digits of each number separately, and then combine those sums in some way such as taking their difference. We test this idea on the provided examples. If a consistent rule emerges, we can apply it to the new pair 82 and 46 with confidence.

Step-by-Step Solution:
Step 1: For 75 $ 26, find the sum of digits of 75: 7 + 5 = 12. Step 2: Find the sum of digits of 26: 2 + 6 = 8. Step 3: Take the difference of these sums: 12 - 8 = 4, which matches the given result. Step 4: For 69 $ 53, sum of digits of 69 is 6 + 9 = 15, and for 53 it is 5 + 3 = 8. Step 5: The difference 15 - 8 = 7 matches the second given result. Step 6: The pattern is therefore clear: a $ b = (sum of digits of a) - (sum of digits of b). Step 7: Now apply the rule to 82 $ 46. Sum of digits of 82 is 8 + 2 = 10. Step 8: Sum of digits of 46 is 4 + 6 = 10. Step 9: The difference is 10 - 10 = 0.

Verification / Alternative check:
Both original equations are perfectly explained by this simple digit sum difference. Alternative ideas, such as subtracting the second full number from the first or adding all digits together, do not consistently give 4 and 7. Because the same neat pattern holds in multiple examples, it is very likely to be the intended rule. Using that rule for the third pair naturally leads to the result 0, which fits the coding logic precisely.

Why Other Options Are Wrong:
Options 62, 56 and 91 might appear if someone misreads the operation as ordinary subtraction or miscalculates with the full numbers. These values do not relate to the digit sums in the way the examples demonstrate. Option 4 repeats the first output but does not follow from the digit sum difference of the third pair, where the sums are equal. Only 0 correctly reflects the rule applied to 82 and 46.

Common Pitfalls:
Students often assume the symbol $ means something like ordinary subtraction or division and do not consider that the digits themselves may be involved. Another frequent mistake is calculating one digit sum correctly but forgetting to subtract in the right order. To avoid this, always keep the pattern simple: compute digit sums separately for each number and then subtract the second sum from the first in a consistent way for every example and the target pair.

Final Answer:
Using the pattern of subtracting digit sums, the value of 82 $ 46 is 0.