Some equations are solved according to a special system. If 5 x 4 x 3 = 70 and 6 x 5 x 4 = 140 under this rule, then what is the correct value of 7 x 6 x 5 in the same pattern?

Introduction / Context:
This question involves a nonstandard way of evaluating expressions with three numbers and the multiplication symbol. The expressions 5 x 4 x 3 and 6 x 5 x 4 do not equal their usual arithmetic products but instead follow a special rule that transforms the product into a different value. Your task is to identify this rule and then apply it to 7 x 6 x 5.

Given Data / Assumptions:
- According to the special system, 5 x 4 x 3 = 70.
- Also, 6 x 5 x 4 = 140.
- We must determine the value of 7 x 6 x 5 under the same rule.
- The pattern should work consistently for both given examples.

Concept / Approach:
The natural starting point is the ordinary product of the three numbers. For 5, 4, and 3, the normal product is 5 * 4 * 3 = 60, which is close to 70. For 6, 5, and 4, the product is 6 * 5 * 4 = 120, which is close to 140. This suggests that a constant extra term, perhaps increasing in a simple way, is being added to the product. We then look at the differences between the standard product and the given result to find a clear pattern.

Step-by-Step Solution:
Step 1: Compute the ordinary product for the first expression: 5 * 4 * 3 = 60. Step 2: Compare with the given result 70. The difference is 70 - 60 = 10. Step 3: For the second expression, compute 6 * 5 * 4 = 120. Step 4: Compare with the given result 140. The difference is 140 - 120 = 20. Step 5: Notice that the additional term increases by 10 when the first number increases from 5 to 6. Step 6: For 7 x 6 x 5, compute the normal product: 7 * 6 * 5 = 210, and then add 30 as the next step in the pattern 10, 20, 30. So 210 + 30 = 240.

Verification / Alternative check:
The pattern of differences 10 and 20 is simple and arithmetic. It is natural to extend it to 30 for the third case, because the first factor increases from 5 to 6 to 7. Any other choice of extra term would break the regular progression. Also, attempts to express 70 and 140 as more complicated functions of the three numbers do not give such a simple and consistent result.

Why Other Options Are Wrong:
- Option 210 equals the plain product 7 * 6 * 5 but ignores the observed extra addition used in the previous equations.
- Option 220 would require an increase of only 10 over 210, which would not match the established pattern of adding 10, then 20, then 30.
- Option 230 would correspond to adding 20, which again fails to continue the increasing sequence of additional terms.

Common Pitfalls:
A common error is to assume that the same number is always added to the product or to try to use overly complex formulas. Another mistake is to overlook the simple arithmetic sequence in the differences between the real products and the coded results. Always examine both the primary arithmetic relationship and the differences to uncover such patterns.

Final Answer:
Using the observed rule of adding 10, 20, and then 30 to successive products, we find that 7 x 6 x 5 = 210 + 30 = 240 under the given system.