The figures 4,5,6,7,8 are written in every possible order. The number of numbers greater than 56000 is 15K then K is equal to ___________________

Answer

To solve the problem, we need to find how many permutations of the digits 4, 5, 6, 7, and 8 result in numbers greater than 56000.

### Step-by-Step Solution:

1. **Identify the digits and the condition**:
We have the digits 4, 5, 6, 7, and 8. We need to find permutations that yield numbers greater than 56000.

2. **Consider the first digit**:
The first digit must be either 5, 6, 7, or 8 to ensure the number is greater than 56000. If the first digit is 4, the number will be less than 56000.

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 digits for the second position are 6, 7, or 8.
- If the second digit is 6, the remaining digits are 4, 7, and 8. The permutations of these three digits are \(3! = 6\).
- If the second digit is 7, the remaining digits are 4, 6, and 8. The permutations of these three digits are \(3! = 6\).
- If the second digit is 8, the remaining digits are 4, 6, and 7. The permutations of these three digits are \(3! = 6\).
- Total for Case 1: \(6 + 6 + 6 = 18\).

4. **Case 2: First digit is 6, 7, or 8**:
- If the first digit is 6, the remaining digits (4, 5, 7, 8) can be arranged in any order. The total permutations are \(4! = 24\).
- If the first digit is 7, the remaining digits (4, 5, 6, 8) can also be arranged in any order. The total permutations are \(4! = 24\).
- If the first digit is 8, the remaining digits (4, 5, 6, 7) can also be arranged in any order. The total permutations are \(4! = 24\).
- Total for Case 2: \(24 + 24 + 24 = 72\).

5. **Combine the results**:
- Total numbers greater than 56000 = Case 1 + Case 2 = \(18 + 72 = 90\).

6. **Relate to the given equation**:
- According to the problem, the total number of numbers greater than 56000 is given as \(15k\).
- Therefore, we set up the equation: \(90 = 15k\).

7. **Solve for k**:
- Dividing both sides by 15 gives \(k = \frac{90}{15} = 6\).

### Final Answer:
Thus, the value of \(k\) is **6**.