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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.
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.
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