To determine the number of even numbers greater than 300 that can be formed using the digits 1, 2, 3, 4, 5 without repetition, we need to consider several factors, including the constraints on even numbers and the range of our target numbers.

Understanding the Criteria
- The number must be **greater than 300**.
- The number must be **even**.
- Use digits **1, 2, 3, 4, 5** without repetition.

Identifying Valid Even Digits
- The even digits available are **2** and **4**.
- Thus, the last digit of the number must be **2** or **4**.

Case Analysis
1. **Case 1: Last Digit is 2**
- Possible digits for the first digit (hundreds place): **3, 4, 5** (to ensure the number is greater than 300).
- If the first digit is:
- **3**: Remaining digits are 1, 4, 5 (2 choices for the second digit).
- **4**: Remaining digits are 1, 3, 5 (2 choices for the second digit).
- **5**: Remaining digits are 1, 3, 4 (2 choices for the second digit).
- Total combinations for this case: **3 * 2 = 6**.
2. **Case 2: Last Digit is 4**
- Possible digits for the first digit (hundreds place): **3, 5** (to ensure the number is greater than 300).
- If the first digit is:
- **3**: Remaining digits are 1, 2, 5 (2 choices for the second digit).
- **5**: Remaining digits are 1, 2, 3 (2 choices for the second digit).
- Total combinations for this case: **2 * 2 = 4**.

Final Calculation
- Total even numbers greater than 300: **6 (from Case 1) + 4 (from Case 2) = 10**.
In conclusion, there are **10 even numbers greater than 300** that can be formed using the digits 1, 2, 3, 4, 5 without repetition.