How many numbers greater than 5000 can be formed with the digits ,3,5,7,8,9 no digit being repeated ?

Options

A. 216
B. 126
C. 512
D. 252

Answer

To solve the problem of how many numbers greater than 5000 can be formed with the digits 3, 5, 7, 8, and 9 without repeating any digit, we can break it down into two cases: forming 4-digit numbers and forming 5-digit numbers.

### Step 1: Count 4-digit numbers greater than 5000

1. **Determine the first digit**: Since we need the number to be greater than 5000, the first digit can only be 5, 7, 8, or 9. This gives us 4 options for the first digit.

2. **Choose the remaining digits**: After selecting the first digit, we have 4 digits left (including 3 and the remaining digits). We need to choose 3 more digits from these remaining 4 digits.

3. **Calculate the arrangements**: The number of ways to arrange these 3 digits is given by the factorial of the number of digits we have left. Thus, the total arrangements for each choice of the first digit is \(3!\).

4. **Total for 4-digit numbers**:
- For each of the 4 choices of the first digit, we can arrange the remaining 3 digits in \(3! = 6\) ways.
- Therefore, the total number of 4-digit numbers greater than 5000 is:
\[ 4 \times 6 = 24 \]

### Step 2: Count 5-digit numbers

1. **Choose the first digit**: For a 5-digit number, any of the 5 digits (3, 5, 7, 8, 9) can be the first digit since all 5-digit numbers will be greater than 5000.

2. **Arrange the remaining digits**: After choosing the first digit, we have 4 remaining digits. The number of ways to arrange these 4 digits is \(4!\).

3. **Total for 5-digit numbers**:
- The total number of 5-digit numbers is:
\[ 5 \times 4! = 5 \times 24 = 120 \]

### Step 3: Combine the totals

Now, we add the totals from both cases:
- Total 4-digit numbers greater than 5000: 24
- Total 5-digit numbers: 120

Thus, the total number of numbers greater than 5000 that can be formed is:
\[ 24 + 120 = 144 \]

### Final Answer
The total number of numbers greater than 5000 that can be formed with the digits 3, 5, 7, 8, and 9 without repeating any digit is **144**.

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