In a special three number code, the expressions 4 x 5 x 2 = 524, 3 x 7 x 2 = 723 and 6 x 8 x 7 = 876 follow the same rearrangement rule. Using this rule, what is the coded value of 9 x 4 x 5?

Introduction / Context:
This question uses a simple rearrangement code involving three numbers. Instead of multiplying or adding them, the three numbers are rearranged to form a three digit number. By comparing the examples, we can infer the positional pattern used and then apply it to the new triple 9, 4 and 5 to determine the coded value.

Given Data / Assumptions:

• 4 x 5 x 2 is coded as 524.
• 3 x 7 x 2 is coded as 723.
• 6 x 8 x 7 is coded as 876.
• We have to find the code for 9 x 4 x 5.
• The symbol x here indicates a special pattern, not multiplication.

Concept / Approach:
The outputs 524, 723 and 876 all contain the same digits as the input numbers but in different positions. This strongly suggests a rearrangement scheme. Our goal is to identify which input number becomes the hundreds digit, which becomes the tens digit and which becomes the units digit. Once we find a consistent mapping that works for all three given equations, we apply that mapping to the new set of numbers.

Step-by-Step Solution:
Step 1: Look at 4 x 5 x 2 = 524. The input numbers are 4, 5 and 2. The result uses digits 5, 2 and 4.Step 2: This suggests the pattern (first, second, third) → (second, third, first). In other words, hundreds digit is the second number, tens digit is the third number and units digit is the first number.Step 3: Check the second example. For 3 x 7 x 2, using the same pattern gives digits 7, 2 and 3, which forms 723. This matches the given result.Step 4: Check the third example. For 6 x 8 x 7, applying the pattern gives 8, 7 and 6, forming 876, which again matches the given value.Step 5: Since the pattern is consistent across all examples, we can confidently apply it to 9 x 4 x 5.Step 6: Here, the first number is 9, the second is 4 and the third is 5. Applying the mapping (first, second, third) → (second, third, first) gives digits 4, 5 and 9.Step 7: Thus, the coded result for 9 x 4 x 5 is 459.

Verification / Alternative check:
If you attempt other rearrangements, such as (third, first, second) or (third, second, first), they will not reproduce all the given examples correctly. Only the mapping second, third, first matches all three coded equations. Rechecking the digits for each case confirms that 459 is the correct and consistent code for 9 x 4 x 5.

Why Other Options Are Wrong:
The option 495 corresponds to a different digit order and would not be produced by the established pattern. Similarly, 549 and 954 represent rearrangements that do not match what happens for the earlier examples. The option 945 also breaks the verified mapping. Therefore, these options are inconsistent with the discovered rule and must be rejected.

Common Pitfalls:
Some students get misled by the multiplication sign and attempt to compute products or sums instead of noticing that the results are simply permutations of the original digits. Others find a rearrangement that works for just one example and do not test it against the others. Always ensure that your pattern is valid for all given examples before using it to solve the final part.

Final Answer:
Using the same rearrangement pattern, the value of 9 x 4 x 5 in this code is 459.