If 19 $ 7 = 312 and 23 $ 9 = 448 according to a certain rule, then using the same rule, what is the value of 31 $ 11 ?

Introduction / Context:
This question features a custom binary operator denoted by the symbol dollar. The operator combines two numbers in a specific way that is not standard addition, subtraction, multiplication or division. We are given two examples of its behaviour and must infer the underlying rule in order to evaluate a third expression involving the same operator. Such questions test pattern recognition and familiarity with square numbers.

Given Data / Assumptions:

• 19 $ 7 = 312.
• 23 $ 9 = 448.
• We must find the value of 31 $ 11.
• The same rule applies consistently across all uses of the dollar operator.

Concept / Approach:
The clue lies in noticing that the results 312 and 448 are quite close to differences of squares. The numbers 19 and 7, for example, suggest checking 19² and 7². By testing whether 19 $ 7 equals 19² minus 7², and similarly for 23 and 9, we can see whether the pattern is difference of squares. If this relationship holds for both examples, we can confidently apply the same formula to 31 and 11.

Step-by-Step Solution:
Step 1: Compute 19² and 7². We have 19² = 361 and 7² = 49. Step 2: Subtract these squares: 361 − 49 = 312. This matches 19 $ 7, so 19 $ 7 is equal to 19² − 7². Step 3: Compute 23² and 9². We have 23² = 529 and 9² = 81. Step 4: Subtract again: 529 − 81 = 448. This matches 23 $ 9 in the question. Step 5: From these two examples we infer that the rule is a $ b = a² − b². Step 6: Apply this rule to 31 $ 11. Compute 31² and 11². Step 7: 31² = 961 and 11² = 121. Step 8: Subtract: 961 − 121 = 840. Step 9: Therefore 31 $ 11 equals 840 under the same rule.

Verification / Alternative check:
We can quickly check that 840 appears among the answer options and that no other simple interpretation of the operator matches both given examples. If we tried sums or products combined with constants, they would not fit both 312 and 448 simultaneously. The difference of squares pattern is exact in both cases and leads to a clear unique value of 840 for the third expression.

Why Other Options Are Wrong:
The alternative values 231, 441 and 641 do not arise naturally from 31 and 11 under the difference of squares rule. For example, 31² is 961 and 11² is 121, and no simple combination of these gives 231, 441 or 641 without contradicting the pattern observed in the first two examples. Choosing any of these values would break the consistency of the rule.

Common Pitfalls:
One common mistake is to look for a rule involving only addition or multiplication without considering squares. Another is to attempt different rules for each example instead of insisting on one rule that works for all. Whenever you see fairly large results alongside moderate inputs, checking sums and differences of squares is a powerful and often successful strategy.

Final Answer:
Using the rule a $ b = a² − b², we find that 31 $ 11 is equal to 840.