To solve the problem of finding the number of integers greater than 7000 that can be formed with the digits 3, 5, 7, 8, and 9 without repeating any digits, 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 7000
1. **Identify valid starting digits**: Since we need a number greater than 7000, the first digit (thousands place) can only be 7, 8, or 9. This gives us 3 choices for the first digit.
- Choices for the first digit: 7, 8, or 9 (3 options)
2. **Choose remaining digits**: After selecting the first digit, we have 4 remaining digits to choose from. We need to fill the hundreds, tens, and units places with these remaining digits.
3. **Calculate arrangements**: The number of ways to arrange the remaining 3 digits is given by the factorial of the number of digits remaining, which is 3! (3 factorial).
- Number of arrangements for the remaining digits = 3! = 6
4. **Total for 4-digit numbers**: Multiply the choices for the first digit by the arrangements of the remaining digits.
\[
\text{Total 4-digit numbers} = 3 \times 6 = 18
\]
### Step 2: Count 5-digit numbers greater than 7000
1. **Choose the first digit**: For a 5-digit number, we can use any of the 5 digits (3, 5, 7, 8, 9) as the first digit since all 5-digit numbers formed will be greater than 7000.
- Choices for the first digit: 5 options (3, 5, 7, 8, 9)
2. **Choose remaining digits**: After selecting the first digit, we have 4 digits left to arrange in the remaining 4 places.
3. **Calculate arrangements**: The number of ways to arrange these 4 remaining digits is given by 4! (4 factorial).
- Number of arrangements for the remaining digits = 4! = 24
4. **Total for 5-digit numbers**: Multiply the choices for the first digit by the arrangements of the remaining digits.
\[
\text{Total 5-digit numbers} = 5 \times 24 = 120
\]
### Step 3: Combine both cases
1. **Add the totals from both cases**:
\[
\text{Total numbers greater than 7000} = \text{Total 4-digit numbers} + \text{Total 5-digit numbers} = 18 + 120 = 138
\]
### Final Answer
The total number of integers greater than 7000 that can be formed with the digits 3, 5, 7, 8, and 9 without repeating any digits is **138**.
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