In a certain arithmetic pattern, it is given that 18 × 12 = 206 and 19 × 22 = 408 (here "×" represents a special operation, not ordinary multiplication). Using the same pattern, what is the value of 23 × 36?

Introduction / Context:
In this question, the multiplication symbol "×" is used to denote a special operation defined by examples rather than normal arithmetic. We are told that 18 × 12 = 206 and 19 × 22 = 408, and asked to compute 23 × 36 under the same rule. This is a typical pattern-recognition puzzle where you must infer the hidden formula from the given data and then apply it consistently.

Given Data / Assumptions:

• 18 × 12 = 206 (by special rule).
• 19 × 22 = 408 (by the same rule).
• We must find 23 × 36.
• The rule should be simple and consistent across all pairs.

Concept / Approach:
We start by checking if the special result is related to ordinary multiplication of the two numbers. Compute 18 × 12 in the usual sense and compare with 206; then do the same for 19 × 22. If a constant difference emerges, that often suggests a simple transformation like (a × b) − k. Once the rule is clear, we apply it to 23 × 36 and choose the matching option.

Step-by-Step Solution:
Step 1: Compute the normal products: 18 × 12 = 216, 19 × 22 = 418. Step 2: Compare with given special results: 216 (normal) → 206 (special), difference = 216 − 206 = 10. 418 (normal) → 408 (special), difference = 418 − 408 = 10. Step 3: Observe that in both cases the special result is exactly 10 less than the ordinary product. Step 4: Hypothesis: a × b (special) = a × b (normal) − 10. Step 5: Now compute 23 × 36 (normal): 23 × 36 = (20 × 36) + (3 × 36) = 720 + 108 = 828. Step 6: Apply the special rule: 828 − 10 = 818.

Verification / Alternative check:
We confirmed that 18 × 12 and 19 × 22 both follow the pattern "normal product minus 10". No other simple pattern, such as adding digits of the numbers or combining them differently, fits both examples as neatly and consistently. Since the special operation must be defined by one stable rule, the formula a × b (special) = a × b (normal) − 10 is the most logical interpretation. Applying it to 23 and 36 yields 818, which matches one of the options exactly.

Why Other Options Are Wrong:
Options A (878), C (794), D (776) and E (806) do not equal 828 − 10 and therefore violate the inferred rule. Choosing any of them would mean the special operation behaves differently for 23 and 36 than it does for the pairs 18, 12 and 19, 22, which contradicts the idea of a single defined operation. Only Option B (818) is consistent with the pattern across all three pairs.

Common Pitfalls:
A frequent mistake is to search for an overly complicated rule involving sums of digits, squares, or concatenations, when a simpler rule based on the normal product plus or minus a constant fits perfectly. Another common pitfall is to check a pattern only against one example and not verify with the second. Always test a hypothesised rule against all given examples before applying it to the target expression.

Final Answer:
Thus, under the special multiplication pattern described, 23 × 36 = 818.