How many four digit numbers can be formed with the digits 0,2,3,5,8,9 if
(i) repetition of digits is allowed ?
(ii ) repetition of digits is not allowed ?

Answer

To solve the problem of how many four-digit numbers can be formed with the digits 0, 2, 3, 5, 8, and 9 under two different conditions (with and without repetition of digits), 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, 2, 3, 5, 8, and 9. This gives us a total of 6 digits.

2. **Determine the first digit**: The first digit of a four-digit number cannot be 0 (otherwise it would be a three-digit number). Therefore, the possible choices for the first digit are 2, 3, 5, 8, or 9. This gives us **5 options** for the first digit.

3. **Determine the remaining digits**: Since repetition is allowed, for the second, third, and fourth digits, we can use any of the 6 digits (including 0). Therefore, we have **6 options** for each of these positions.

4. **Calculate the total combinations**: The total number of four-digit numbers can be calculated by multiplying the number of choices for each digit:
\[ \text{Total} = (\text{Choices for 1st digit}) \times (\text{Choices for 2nd digit}) \times (\text{Choices for 3rd digit}) \times (\text{Choices for 4th digit}) \]
\[ \text{Total} = 5 \times 6 \times 6 \times 6 = 5 \times 216 = 1080 \]

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

1. **Identify the digits available**: Again, we have the digits 0, 2, 3, 5, 8, and 9 (6 digits).

2. **Determine the first digit**: The first digit cannot be 0, so we again have 5 options (2, 3, 5, 8, or 9).

3. **Determine the second digit**: After choosing the first digit, we cannot use that digit again. If we filled the first digit with one of the 5 options, we have 5 remaining digits (including 0) left to choose from for the second digit.

4. **Determine the third digit**: After choosing the first and second digits, we have 4 remaining digits to choose from for the third digit.

5. **Determine the fourth digit**: After choosing the first, second, and third digits, we have 3 remaining digits to choose from for the fourth digit.

6. **Calculate the total combinations**: The total number of four-digit numbers can be calculated by multiplying the number of choices for each digit:
\[ \text{Total} = (\text{Choices for 1st digit}) \times (\text{Choices for 2nd digit}) \times (\text{Choices for 3rd digit}) \times (\text{Choices for 4th digit}) \]
\[ \text{Total} = 5 \times 5 \times 4 \times 3 = 300 \]

### Final Answers
- (i) The total number of four-digit numbers with repetition allowed is **1080**.
- (ii) The total number of four-digit numbers without repetition is **300**.