How many numbers of 5 digits can be formed with the digits 1,3,0,2,4,8 if
(i) repetition is not allowed?
(ii) repetition is allowed?

Answer

To solve the problem of how many 5-digit numbers can be formed with the digits 1, 3, 0, 2, 4, and 8 under two different conditions (repetition not allowed and repetition allowed), we can break it down into two parts.

### Part (i): Repetition Not Allowed

1. **Identify the digits available**: The digits we can use are 1, 3, 0, 2, 4, and 8. This gives us a total of 6 digits.

2. **Determine the first digit**: The first digit of a 5-digit number cannot be 0 (as it would then be a 4-digit number). Therefore, the options for the first digit are 1, 3, 2, 4, or 8. This gives us 5 choices.

3. **Determine the remaining digits**:
- After choosing the first digit, we have used one digit, leaving us with 5 remaining digits.
- The second digit can be any of the remaining 5 digits (including 0).
- The third digit can be any of the remaining 4 digits.
- The fourth digit can be any of the remaining 3 digits.
- The fifth digit can be any of the remaining 2 digits.

4. **Calculate the total combinations**:
- The total number of 5-digit numbers can be calculated as:
\[ \text{Total} = 5 \times 5 \times 4 \times 3 \times 2 \]
- Performing the calculations:
\[ = 5 \times 5 = 25 \]
\[ = 25 \times 4 = 100 \]
\[ = 100 \times 3 = 300 \]
\[ = 300 \times 2 = 600 \]

Thus, the total number of 5-digit numbers that can be formed without repetition is **600**.

### Part (ii): Repetition Allowed

1. **Identify the digits available**: The digits we can use are still 1, 3, 0, 2, 4, and 8 (6 digits).

2. **Determine the first digit**: As before, the first digit cannot be 0. Therefore, we have 5 choices for the first digit (1, 3, 2, 4, or 8).

3. **Determine the remaining digits**: Since repetition is allowed:
- The second digit can be any of the 6 digits (1, 3, 0, 2, 4, or 8).
- The third digit can also be any of the 6 digits.
- The fourth digit can also be any of the 6 digits.
- The fifth digit can also be any of the 6 digits.

4. **Calculate the total combinations**:
- The total number of 5-digit numbers can be calculated as:
\[ \text{Total} = 5 \times 6 \times 6 \times 6 \times 6 \]
- This can also be expressed as:
\[ = 5 \times 6^4 \]
- Calculating \(6^4\):
\[ 6^4 = 1296 \]
- Therefore, the total is:
\[ = 5 \times 1296 = 6480 \]

Thus, the total number of 5-digit numbers that can be formed with repetition allowed is **6480**.

### Final Answers:
- For part (i): **600** numbers can be formed without repetition.
- For part (ii): **6480** numbers can be formed with repetition allowed.