The number of numbers greater than 56000 that can be formed by using the digits 4,5,6,7,8, no digit being repeated in any number is

Options

A. 18
B. 36
C. 72
D. 90

Answer

To solve the problem of finding the number of numbers greater than 56000 that can be formed using the digits 4, 5, 6, 7, and 8 without repeating any digits, we can break it down into two cases based on the first digit of the number.

### Step-by-Step Solution:

1. **Identify the digits available**: The digits we can use are 4, 5, 6, 7, and 8.

2. **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. This gives us 3 options for the second digit.
- After choosing the first two digits, we have 3 remaining digits left to fill the last three positions. The number of ways to arrange these 3 digits is given by \(3!\) (3 factorial).
- Calculation:
- Number of ways to choose the second digit: 3
- Number of ways to arrange the remaining 3 digits: \(3! = 6\)
- Total for this case: \(1 \times 3 \times 6 = 18\)

3. **Case 2: First digit is greater than 5**
- The possible first digits in this case are 6, 7, or 8. This gives us 3 options for the first digit.
- After choosing the first digit, we have 4 remaining digits to fill the last four positions. The number of ways to arrange these 4 digits is given by \(4!\) (4 factorial).
- Calculation:
- Number of ways to choose the first digit: 3
- Number of ways to arrange the remaining 4 digits: \(4! = 24\)
- Total for this case: \(3 \times 24 = 72\)

4. **Combine both cases**:
- Total numbers greater than 56000 = Total from Case 1 + Total from Case 2
- Total = \(18 + 72 = 90\)

### Final Answer:
The total number of numbers greater than 56000 that can be formed is **90**.