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

Let's solve the problem step by step.

### Given:
Digits available: 1, 3, 0, 2, 4, 8

### Part (i): Repetition is not allowed

1. **Determine the first digit**:
- Since we are forming a 5-digit number, the first digit cannot be 0 (as it would not be a 5-digit number).
- The possible choices for the first digit are: 1, 2, 3, 4, 8.
- Therefore, we have **5 choices** for the first digit.

2. **Determine the second digit**:
- After choosing the first digit, we have 5 digits left (including 0).
- Thus, we have **5 choices** for the second digit.

3. **Determine the third digit**:
- After choosing the first and second digits, we have 4 digits remaining.
- Therefore, we have **4 choices** for the third digit.

4. **Determine the fourth digit**:
- After choosing the first three digits, we have 3 digits remaining.
- Thus, we have **3 choices** for the fourth digit.

5. **Determine the fifth digit**:
- After choosing the first four digits, we have 2 digits remaining.
- Therefore, we have **2 choices** for the fifth digit.

6. **Calculate the total number of combinations**:
- The total number of 5-digit numbers that can be formed is calculated by multiplying the choices together:
\[ \text{Total} = 5 \times 5 \times 4 \times 3 \times 2 \]
\[ = 600 \]

### Part (ii): Repetition is allowed

1. **Determine the first digit**:
- The first digit cannot be 0.
- The possible choices for the first digit are: 1, 2, 3, 4, 8.
- Therefore, we have **5 choices** for the first digit.

2. **Determine the remaining digits**:
- For the second, third, fourth, and fifth digits, since repetition is allowed, we can use any of the 6 digits (including 0).
- Therefore, we have **6 choices** for each of the remaining 4 digits.

3. **Calculate the total number of combinations**:
- The total number of 5-digit numbers that can be formed is calculated as follows:
\[ \text{Total} = 5 \times 6 \times 6 \times 6 \times 6 \]
\[ = 5 \times 6^4 \]
\[ = 5 \times 1296 = 6480 \]

### Final Answers:
- (i) The total number of 5-digit numbers that can be formed without repetition is **600**.
- (ii) The total number of 5-digit numbers that can be formed with repetition is **6480**.