Let's solve the problem step by step:
### Information Given:
- Digits available: 0, 1, 2, 3, 4, 5, 7, 8
- The number must be a two-digit odd number.
- No digit may be repeated in any number.
### Requirements for Two-Digit Odd Numbers:
1. The tens place must be occupied by any digit from the set except 0.
2. The units place must be occupied by any digit from the set that makes the number odd. Hence, the units digit should be one of the odd digits: 1, 3, 5, or 7.
### Steps to Find the Two-Digit Odd Numbers:
1. Identify potential tens place digits: The possible digits for the tens place are 1, 2, 3, 4, 5, 7, and 8. We exclude 0 because a two-digit number cannot start with 0.
2. Identify potential units place digits: The possible digits for the units place that make the number odd are 1, 3, 5, and 7.
3. Form combinations: Now, combine each tens place digit with each valid units place digit, ensuring no digit is repeated within the same number.
#### Form Combinations:
We will combine each tens place digit with each valid unit digit if they are different:
- For tens digit 1: valid units are 3, 5, 7 (3 combinations)
- For tens digit 2: valid units are 1, 3, 5, 7 (4 combinations)
- For tens digit 3: valid units are 1, 5, 7 (3 combinations)
- For tens digit 4: valid units are 1, 3, 5, 7 (4 combinations)
- For tens digit 5: valid units are 1, 3, 7 (3 combinations)
- For tens digit 7: valid units are 1, 3, 5 (3 combinations)
- For tens digit 8: valid units are 1, 3, 5, 7 (4 combinations)
Now we sum up all the valid combinations:
- Tens digit 1: 3 combinations
- Tens digit 2: 4 combinations
- Tens digit 3: 3 combinations
- Tens digit 4: 4 combinations
- Tens digit 5: 3 combinations
- Tens digit 7: 3 combinations
- Tens digit 8: 4 combinations
#### Total Combinations:
Adding them up:
[tex]\[ 3 + 4 + 3 + 4 + 3 + 3 + 4 = 24 \][/tex]
### Conclusion:
The total number of valid two-digit odd numbers that can be formed is 24.
