In a three number coding system, 5 x 4 x 0 is written as 405 and 3 x 2 x 8 is written as 283. Using the same coding pattern, what should be the value of 1 x 7 x 6?

Introduction / Context:
This verbal reasoning question presents a simple three number code where expressions like 5 x 4 x 0 and 3 x 2 x 8 are converted into three digit numbers. The symbol x does not indicate multiplication but an arrangement of digits. By inspecting the examples, we must infer the order in which the three numbers are written to form the coded output.

Given Data / Assumptions:

• 5 x 4 x 0 is coded as 405.
• 3 x 2 x 8 is coded as 283.
• We are asked to find the coded value of 1 x 7 x 6.
• The three digits in the result must come from the three original numbers.
• The same arrangement rule is applied consistently in all cases.

Concept / Approach:
The outputs 405 and 283 clearly use the same digits as the inputs but in different positions. This indicates that the code is a permutation of the original numbers. We must determine which input number becomes the hundreds digit, which becomes the tens digit and which becomes the units digit by analyzing the examples, and then apply this pattern to 1, 7 and 6.

Step-by-Step Solution:
Step 1: Look at 5 x 4 x 0 = 405. The inputs are 5, 4 and 0, while the result 405 uses the digits 4, 0 and 5.Step 2: This suggests the pattern (first, second, third) → (second, third, first), where the hundreds digit is the second number, the tens digit is the third number and the units digit is the first number.Step 3: Check the second example. For 3 x 2 x 8, the digits are 3, 2 and 8. Using the pattern second, third, first, we get 2, 8 and 3, which forms 283. This matches the given code.Step 4: Since the pattern holds for both examples, we can confidently apply it to 1 x 7 x 6.Step 5: For 1 x 7 x 6, the first number is 1, the second is 7 and the third is 6.Step 6: According to the mapping, hundreds digit is the second number (7), tens digit is the third number (6) and units digit is the first number (1), giving 7, 6 and 1.Step 7: Therefore, 1 x 7 x 6 should be coded as 761.

Verification / Alternative check:
Trying other permutations such as third, first, second or third, second, first on the examples would not reproduce both 405 and 283 correctly. Only the second, third, first order consistently matches the provided coded results. Reapplying this confirmed rule to 1, 7 and 6 gives 761, so the answer is reliable.

Why Other Options Are Wrong:
The options 617, 716, 167 and 671 correspond to different permutations of 1, 7 and 6. None of those follow the discovered pattern that places the second number first, the third number second and the first number last. Therefore they cannot match the coding rule that was used for the given examples.

Common Pitfalls:
Some students treat x as multiplication and attempt to compute products before noticing that the outputs are simply rearrangements. Others find a pattern that works for only one example and do not check the others. Always ensure that your inferred rule fits every given equation, not just one, before applying it to find the final answer.

Final Answer:
Using the same rearrangement pattern as in the examples, the coded value of 1 x 7 x 6 is 761.