The figures 4, 5, 6, 7, 8 are written in every possible order. The number of numbers greater than 56000 is __________

Answer

To find the number of permutations of the digits 4, 5, 6, 7, and 8 that are greater than 56000, we can follow these steps:

### 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 need to consider the possible first digits:
- If the first digit is 5, the number must be greater than 56000.
- If the first digit is 6, 7, or 8, all numbers will be greater than 56000.

3. **Case 1: First Digit is 5**:
- The next digit must be either 6, 7, or 8 to ensure the number is greater than 56000.
- **Sub-case 1.1: Second Digit is 6**:
- Remaining digits: 4, 7, 8.
- Total permutations = 3! = 6.
- **Sub-case 1.2: Second Digit is 7**:
- Remaining digits: 4, 6, 8.
- Total permutations = 3! = 6.
- **Sub-case 1.3: Second Digit is 8**:
- Remaining digits: 4, 6, 7.
- Total permutations = 3! = 6.
- Total for Case 1 = 6 + 6 + 6 = 18.

4. **Case 2: First Digit is 6**:
- Remaining digits: 4, 5, 7, 8.
- Total permutations = 4! = 24.

5. **Case 3: First Digit is 7**:
- Remaining digits: 4, 5, 6, 8.
- Total permutations = 4! = 24.

6. **Case 4: First Digit is 8**:
- Remaining digits: 4, 5, 6, 7.
- Total permutations = 4! = 24.

7. **Total Count**:
- Total numbers greater than 56000 = Case 1 + Case 2 + Case 3 + Case 4
- Total = 18 (from Case 1) + 24 (from Case 2) + 24 (from Case 3) + 24 (from Case 4) = 90.

### Final Answer:
The number of numbers greater than 56000 is **90**.