To solve the problem of finding how many integers greater than 7000 can be formed with the digits 3, 5, 7, 8, and 9 without repeating any digits, we will break it down into two cases: forming 4-digit numbers and forming 5-digit numbers.
### Step 1: Count 4-digit numbers greater than 7000
For a 4-digit number to be greater than 7000, the first digit must be either 7, 8, or 9.
#### Case A: First digit is 7
- Remaining digits: 3, 5, 8, 9 (4 options)
- The second digit can be chosen in 4 ways.
- The third digit can be chosen in 3 ways.
- The fourth digit can be chosen in 2 ways.
Total combinations for this case:
\[
4 \times 3 \times 2 = 24
\]
#### Case B: First digit is 8
- Remaining digits: 3, 5, 7, 9 (4 options)
- The second digit can be chosen in 4 ways.
- The third digit can be chosen in 3 ways.
- The fourth digit can be chosen in 2 ways.
Total combinations for this case:
\[
4 \times 3 \times 2 = 24
\]
#### Case C: First digit is 9
- Remaining digits: 3, 5, 7, 8 (4 options)
- The second digit can be chosen in 4 ways.
- The third digit can be chosen in 3 ways.
- The fourth digit can be chosen in 2 ways.
Total combinations for this case:
\[
4 \times 3 \times 2 = 24
\]
### Total for 4-digit numbers:
Adding all cases together:
\[
24 + 24 + 24 = 72
\]
### Step 2: Count 5-digit numbers greater than 7000
All 5-digit numbers formed with the digits 3, 5, 7, 8, and 9 will be greater than 7000 since they are 5 digits long.
- Total digits available: 5 (3, 5, 7, 8, 9)
- The first digit can be any of the 5 digits.
- The second digit can be chosen in 4 ways.
- The third digit can be chosen in 3 ways.
- The fourth digit can be chosen in 2 ways.
- The fifth digit can be chosen in 1 way.
Total combinations for 5-digit numbers:
\[
5 \times 4 \times 3 \times 2 \times 1 = 120
\]
### Step 3: Calculate total combinations (m)
Now, we combine the totals from both cases:
\[
m = 72 + 120 = 192
\]
### Step 4: Find \( \frac{m}{100} \)
To find the final answer:
\[
\frac{m}{100} = \frac{192}{100} = 1.92
\]
### Final Answer:
The value of \( \frac{m}{100} \) is \( 1.92 \).
---
