In this coding puzzle, three digit expressions are rewritten by rearranging their digits. It is given that 6 x 2 x 9 is written as 269 and 8 x 7 x 1 is written as 781. On the same basis, how should 4 x 1 x 3 be written?

Introduction / Context:
This question involves a simple digit rearrangement code. The numbers 6, 2 and 9 are written with multiplication signs between them as 6 x 2 x 9, but the answer 269 is just a three digit number whose digits are drawn from 6, 2 and 9. The second example, 8 x 7 x 1 = 781, follows the same pattern. You need to identify how the digits are being rearranged and then apply that to 4 x 1 x 3.

Given Data / Assumptions:

• 6 x 2 x 9 is coded as 269.
• 8 x 7 x 1 is coded as 781.
• The multiplication signs are not used for actual multiplication; they simply separate the three input digits.
• The code rearranges the three digits into a new three digit number.
• We must determine the rearrangement for 4 x 1 x 3.

Concept / Approach:
The first step is to examine the relation between the ordered triple (6, 2, 9) and the digits of 269. Observing that the coded form starts with the middle number, followed by the first number and then the last number, suggests a (second, first, third) pattern. Confirming this with the second example (8, 7, 1) and 781 will show whether the pattern is consistent. Once it is confirmed, we can apply the same rearrangement to the new triple (4, 1, 3).

Step-by-Step Solution:
Step 1: For 6 x 2 x 9 → 269, note the original digits in order: first 6, second 2, third 9. Step 2: The coded number 269 has digits 2, 6 and 9 in that order. Step 3: So the pattern is: take the second digit first, then the first digit, then the third digit, yielding (second, first, third). Step 4: Check this rule with the second example, 8 x 7 x 1 → 781. Original digits: first 8, second 7, third 1. Applying (second, first, third): 7, 8, 1 → 781, which matches the coded result. Step 5: Now apply the same rearrangement to 4 x 1 x 3. Original digits: first 4, second 1, third 3. Rearranged as (second, first, third): 1, 4, 3 → 143.

Verification / Alternative check:
Since the same (second, first, third) rule explains both given examples perfectly, there is no need to search for a more complicated pattern. Any other rearrangement, such as (first, third, second) or (third, second, first), would fail to match at least one of the examples. Thus, applying the (second, first, third) pattern to every triple is consistent and simple. For 4, 1, 3, this gives 143, which matches option D exactly.

Why Other Options Are Wrong:
Options 431, 413, 341 and 314 correspond to different permutations of the digits 4, 1 and 3, but none of these follows the consistent (second, first, third) mapping that the given examples confirm. Choosing any of these would require a different rearrangement rule that would not hold for both 6, 2, 9 and 8, 7, 1. Therefore, they cannot represent the intended coding system.

Common Pitfalls:
A common mistake is to try to interpret the x symbols as actual multiplication, which leads to numerical products instead of digit rearrangements and does not fit the outputs. Another error is to choose a permutation that works for only one example but not both. Always use all given examples to identify a single consistent rule, then apply that rule to the new case.

Final Answer:
Using the same rearrangement pattern, 4 x 1 x 3 is written as 143.