In the following pattern based system, some three number expressions are transformed in a consistent way: 7 x 6 x 8 becomes 678, 8 x 9 x 7 becomes 987, and 6 x 5 x 7 becomes 567. Using the same pattern, what number will replace the question mark in 5 x 4 x 6 = ?

Introduction / Context:
This is a pattern recognition question where three numbers linked by a multiplication sign are transformed into a three digit number. The operation is not actual multiplication, but a simple rearrangement of the digits. Your job is to understand how the digits from the left side are rearranged to form the number on the right side, and then apply the same rule to the new triple 5, 4, and 6.

Given Data / Assumptions:

• 7 x 6 x 8 is written as 678. • 8 x 9 x 7 is written as 987. • 6 x 5 x 7 is written as 567. • Target expression: 5 x 4 x 6. • The symbol x here indicates a pattern separator, not real multiplication.

Concept / Approach:
These examples suggest that the three numbers on the left are simply rearranged to form the three digit number on the right. By comparing each triple with its result, we can see how the positions change. Once we find a consistent rule for the order in which the numbers appear, we apply it to the new triple 5, 4, and 6. There is no actual arithmetic computation, only positional rearrangement.

Step-by-Step Solution:
Step 1: Represent each triple as (a, b, c). Step 2: For 7 x 6 x 8, we have a = 7, b = 6, c = 8, and the result is 678. This corresponds to the order (b, a, c) = (6, 7, 8). Step 3: For 8 x 9 x 7, a = 8, b = 9, c = 7, and the result is 987. Again, this matches (b, a, c) = (9, 8, 7). Step 4: For 6 x 5 x 7, a = 6, b = 5, c = 7, and the result 567 corresponds to (b, a, c) = (5, 6, 7). Step 5: Thus the rule is clear: the three digit result is formed by writing b first, a second, and c third. Step 6: Apply this to 5 x 4 x 6. Here a = 5, b = 4, c = 6. Step 7: According to the rule (b, a, c), we write 4, 5, 6, which gives 456.

Verification / Alternative check:
You can check that this ordering rule works for every given example. Any other fixed ordering of a, b, and c, such as (a, b, c) or (c, b, a), would fail for at least one of the supplied cases. Since (b, a, c) is consistent in all three examples and also produces a meaningful result for 5, 4, and 6, we can be confident that 456 is the intended answer.

Why Other Options Are Wrong:
Option 564 corresponds to (a, c, b) and does not match the rule used in the examples. Option 645 corresponds loosely to (c, a, b) and is again inconsistent with the given pattern. Option 654 corresponds to (c, b, a) and fails to reproduce the sample mappings. Option 546 corresponds to (a, b, c), the original order, but the examples clearly show that the result is not a simple copy of the triple.

Common Pitfalls:
Some learners try to interpret the expression as actual multiplication or perform operations like a + b + c or a × b × c, which does not match the examples. Others may guess based on ascending or descending order alone without checking all given cases. Always verify a hypothesized pattern against every example provided before using it to find the missing term.

Final Answer:
Following the rule that the three digit number is formed in the order (b, a, c), the expression 5 x 4 x 6 is written as 456.