To determine how many two-digit odd numbers can be made using the digits 0, 1, 2, 3, 4, 5, 7, and 8, with each digit being used at most once in a number, we can follow these steps:
1. Identify the requirements for the two-digit number:
- It must be a two-digit number, meaning the tens place cannot be 0.
- It must be an odd number, meaning the ones place must be an odd digit.
2. List the available digits for each place:
- Digits available: 0, 1, 2, 3, 4, 5, 7, 8.
- Odd digits available (for the units place): 1, 3, 5, 7.
3. Iterate through the digits and count valid combinations:
For each valid odd digit in the units place:
- Units Digit = 1:
The tens digit can be any of 2, 3, 4, 5, 7, or 8 (0 and 1 are not allowed here).
Possible tens digits: 2, 3, 4, 5, 7, 8.
This gives us 6 combinations.
- Units Digit = 3:
The tens digit can be any of 2, 4, 5, 7, 8, or 1 (0 and 3 are not allowed here).
Possible tens digits: 2, 4, 5, 7, 8, 1.
This gives us 6 combinations.
- Units Digit = 5:
The tens digit can be any of 2, 3, 4, 7, 8, or 1 (0 and 5 are not allowed here).
Possible tens digits: 2, 3, 4, 7, 8, 1.
This gives us 6 combinations.
- Units Digit = 7:
The tens digit can be any of 2, 3, 4, 5, 8, or 1 (0 and 7 are not allowed here).
Possible tens digits: 2, 3, 4, 5, 8, 1.
This gives us 6 combinations.
4. Sum of all valid combinations:
- Total combinations = 6 (for 1) + 6 (for 3) + 6 (for 5) + 6 (for 7) = 24.
Therefore, the number of two-digit odd numbers that can be made using the digits 0, 1, 2, 3, 4, 5, 7, 8, with each digit used at most once, is 24.
