Step 1 : Understanding the Question
This is a logical reasoning puzzle based on a numeric matrix. In such problems, the goal is to discover a consistent mathematical rule or pattern that links the numbers in each row or column. In this specific $3 \times 3$ grid, the middle column appears to be a result derived from the operation of the outer numbers in the first and third columns. The solution requires testing various arithmetic operations (addition, multiplication, powers) until a rule holds true for all rows.
Step 2 : Key Formulas and approach
The strategy involves identifying a relationship between $C_1$ (Column 1) and $C_3$ (Column 3) that produces $C_2$ (Column 2). By observing that the middle numbers (210, 156) are slightly less than the product of the outer numbers ($16 \times 14 = 224$ and $14 \times 12 = 168$), we can hypothesize a subtraction-based logic.
Key Formulas tested:
1. $C_2 = (C_1 \times C_3) - C_3$
2. $C_2 = (C_1 - 1) \times C_3$
Step 3 : Detailed Explanation
Analyzing Row 1: Let's apply the product rule. $16 \times 14 = 224$. To reach the center value of 210, we must subtract 14. Thus, $(16 \times 14) - 14 = 210$. This suggests the pattern: $(C_1 \times C_3) - C_3 = C_2$.
Testing Row 2: We verify the pattern with the second row. $14 \times 12 = 168$. Subtracting the third column value (12) gives $168 - 12 = 156$. The rule is consistent.
Alternative Logic: We can also express this as $C_3(C_1 - 1) = C_2$. In Row 1, $14 \times (16 - 1) = 14 \times 15 = 210$. In Row 2, $12 \times (14 - 1) = 12 \times 13 = 156$. Both logics lead to the same result.
Applying to Row 3: Now, we use the confirmed rule on the final row to find the missing value (?). Using $C_1 = 12$ and $C_3 = 10$, we calculate: $(12 \times 10) - 10$.
Final Calculation: $120 - 10 = 110$. Or using the alternative method: $(12 - 1) \times 10 = 11 \times 10 = 110$.
Step 4 : Final Answer
The missing number in the matrix is 110, which matches option (C).