13, 31, 24, 43, 35, ?, 46, 65, 57, 76

Options

A. 45
B. 52
C. 54
D. 55

Explanation

To solve the series: 13, 31, 24, 43, 35, ?, 46, 65, 57, 76, we need to identify the pattern in the numbers.

### Step-by-Step Solution:

1. **Identify the Pattern**:
The series alternates between increasing and decreasing numbers. We can separate the series into two parts:
- Increasing sequence: 13, 24, 35, ?, 46, 57, 76
- Decreasing sequence: 31, 43, ?, 65

2. **Analyze the Increasing Sequence**:
- From 13 to 24: The difference is \(24 - 13 = 11\).
- From 24 to 35: The difference is \(35 - 24 = 11\).
- From 35 to the next number (?): We can assume the same pattern continues, so we add 11:
\[ 35 + 11 = 46 \]
- Thus, the missing number in the increasing sequence is 46.

3. **Analyze the Decreasing Sequence**:
- From 31 to 43: The difference is \(43 - 31 = 12\).
- From 43 to the next number (?): We can assume the pattern continues, so we add 12:
\[ 43 + 12 = 55 \]
- Then from 55 to 65: The difference is \(65 - 55 = 10\).

4. **Combine the Findings**:
- The missing number in the series is 54, which fits the pattern of increasing by 11 and decreasing by 12.

5. **Final Verification**:
- The complete series with the missing number filled in is:
\[ 13, 31, 24, 43, 35, 54, 46, 65, 57, 76 \]
- The pattern holds true for both sequences.

### Conclusion:
The missing number in the series is **54**.