To solve the problem, we need to find how many numbers can be formed using the digits 4, 5, 6, 7, and 8 that are greater than 56000.
### Step-by-Step Solution:
1. **Identify the digits**: The digits we have are 4, 5, 6, 7, and 8.
2. **Determine the first digit**: Since we want numbers greater than 56000, we can start by considering the possible first digits:
- The first digit can be 5, 6, 7, or 8.
3. **Case 1: First digit is 5**
- If the first digit is 5, the second digit must be greater than 6 to ensure the number is greater than 56000.
- The possible second digits are 6, 7, or 8.
- **Sub-case 1.1: Second digit is 6**
- The remaining digits are 4, 7, and 8. We can arrange these in 3! = 6 ways.
- **Sub-case 1.2: Second digit is 7**
- The remaining digits are 4, 6, and 8. We can arrange these in 3! = 6 ways.
- **Sub-case 1.3: Second digit is 8**
- The remaining digits are 4, 6, and 7. We can arrange these in 3! = 6 ways.
- Total for Case 1: 6 + 6 + 6 = 18 ways.
4. **Case 2: First digit is 6**
- Any arrangement of the remaining digits (4, 5, 7, 8) will yield a number greater than 56000.
- The number of arrangements is 4! = 24 ways.
5. **Case 3: First digit is 7**
- Any arrangement of the remaining digits (4, 5, 6, 8) will yield a number greater than 56000.
- The number of arrangements is 4! = 24 ways.
6. **Case 4: First digit is 8**
- Any arrangement of the remaining digits (4, 5, 6, 7) will yield a number greater than 56000.
- The number of arrangements is 4! = 24 ways.
7. **Total Numbers Greater than 56000**:
- Total = Case 1 + Case 2 + Case 3 + Case 4
- Total = 18 + 24 + 24 + 24 = 90 ways.
8. **Relate to the given condition**:
- We know from the problem statement that the total number of numbers greater than 56000 is given as 15K.
- Therefore, we set up the equation: 15K = 90.
9. **Solve for K**:
- K = 90 / 15 = 6.
### Final Answer:
K is equal to **6**.
