How many numbers of 4 digits can be formed with the digits 0,1,3,4,6 if
(i) repetition of digits is allowed?
(ii) repetition of digits is not allowed?

Answer

To solve the problem of how many 4-digit numbers can be formed with the digits 0, 1, 3, 4, and 6 under two different conditions (repetition allowed and repetition not allowed), we will break it down step by step.

### Part (i): Repetition of digits is allowed

1. **Identify the digits available**: The digits we can use are 0, 1, 3, 4, and 6. This gives us a total of 5 digits.

2. **Determine the first digit**: The first digit of a 4-digit number cannot be 0 (otherwise it would be a 3-digit number). Therefore, we have 4 options for the first digit: 1, 3, 4, or 6.

3. **Determine the remaining digits**: For the second, third, and fourth digits, we can use any of the 5 available digits (including 0) since repetition is allowed.

4. **Calculate the total combinations**:
- Choices for the first digit: 4
- Choices for the second digit: 5
- Choices for the third digit: 5
- Choices for the fourth digit: 5

The total number of 4-digit numbers is calculated as:
\[ \text{Total} = 4 \times 5 \times 5 \times 5 = 4 \times 125 = 500 \]

### Part (ii): Repetition of digits is not allowed

1. **Identify the digits available**: Again, the digits are 0, 1, 3, 4, and 6.

2. **Determine the first digit**: The first digit cannot be 0, so we have 4 options (1, 3, 4, or 6).

3. **Determine the second digit**: After choosing the first digit, we cannot use it again. If the first digit is one of the 4 options, we are left with 4 digits (including 0) for the second digit.

4. **Determine the third digit**: After choosing the first and second digits, we cannot use either of them again. This leaves us with 3 digits to choose from for the third digit.

5. **Determine the fourth digit**: After choosing the first three digits, we cannot use any of them again. This leaves us with 2 digits to choose from for the fourth digit.

6. **Calculate the total combinations**:
- Choices for the first digit: 4
- Choices for the second digit: 4 (since one digit is already used)
- Choices for the third digit: 3 (since two digits are already used)
- Choices for the fourth digit: 2 (since three digits are already used)

The total number of 4-digit numbers is calculated as:
\[ \text{Total} = 4 \times 4 \times 3 \times 2 = 96 \]

### Final Answers
- (i) The total number of 4-digit numbers when repetition is allowed is **500**.
- (ii) The total number of 4-digit numbers when repetition is not allowed is **96**.