The missing number in the following series is: 15 (105) 14, 13 (?) 12. Choose the correct alternative that fits the same pattern as the first bracketed number.

Introduction / Context:
This question checks your understanding of number relationships using brackets. You are given one complete pattern, 15 (105) 14, and then asked to apply the same hidden rule to a second case, 13 (?) 12. Such questions appear regularly in reasoning sections to test how quickly you can discover and transfer a numerical rule.

Given Data / Assumptions:

- First pattern: 15 (105) 14.- Second pattern: 13 (?) 12.- The same rule must connect the outer numbers to the middle number in both cases.

Concept / Approach:
When numbers appear outside and inside brackets like this, a common idea is that the middle number is derived from combining the two outer numbers using multiplication, addition, or a fraction of their product. Here, the outer numbers in the first pattern are 15 and 14, so it is natural to try operations involving 15 * 14 or 15 + 14 and see what yields 105.

Step-by-Step Solution:
Step 1: Multiply the outer numbers in the first pattern: 15 * 14 = 210.Step 2: Observe that 105 is exactly half of 210, so 105 = (15 * 14) / 2.Step 3: Thus, the rule is: middle number = (first outer number * second outer number) / 2.Step 4: Apply the same rule to the second pattern. Multiply the outer numbers: 13 * 12 = 156.Step 5: Divide by 2 to match the discovered pattern: 156 / 2 = 78.Step 6: Therefore, the missing number in the second pattern is 78.

Verification / Alternative check:
Check the rule once more for the first pattern: (15 * 14) / 2 = 210 / 2 = 105, which is correct. Then check the second pattern: (13 * 12) / 2 = 156 / 2 = 78, which matches our result. Since the same relationship holds for both cases, the rule is consistent and reliable.

Why Other Options Are Wrong:
Option 91 does not equal (13 * 12) / 2, because 91 is not half of 156. Option 65 is smaller than half of 156, and option 68 is also not equal to 78. None of these values can be obtained using the same rule that produced 105 from 15 and 14, so they break the pattern.

Common Pitfalls:
Students sometimes try arbitrary operations like adding a fixed number or squaring individual terms without checking whether the same idea works consistently across all given patterns. Another mistake is to focus on only one of the outer numbers rather than using both together. Always verify that the rule you propose explains every example provided in the question.

Final Answer:
The missing number that completes the second pattern is 78.