How many 4-digit numbers can be formed with the digits 0, 2, 3, 4, 5 when a digit may be repeated any number of times in any arrangement?

Explanation

To solve the problem of how many 4-digit numbers can be formed using the digits 0, 2, 3, 4, and 5 with repetition allowed, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the digits available**: The digits we can use are 0, 2, 3, 4, and 5.

2. **Determine the first digit**: Since we are forming a 4-digit number, the first digit cannot be 0 (as it would make it a 3-digit number). Therefore, the possible choices for the first digit are 2, 3, 4, and 5. This gives us 4 options for the first digit.

**Hint**: Remember that the first digit of a multi-digit number cannot be zero.

3. **Determine the remaining digits**: For the second, third, and fourth digits, we can use any of the 5 digits (including 0) since repetition is allowed. Thus, we have 5 options for each of these positions.

**Hint**: Since repetition is allowed, all digits can be used for the remaining positions.

4. **Calculate the total combinations**:
- For the first digit, we have 4 choices.
- For the second digit, we have 5 choices.
- For the third digit, we have 5 choices.
- For the fourth digit, we have 5 choices.

Therefore, the total number of 4-digit numbers can be calculated as:
\[ \text{Total Numbers} = (\text{Choices for 1st digit}) \times (\text{Choices for 2nd digit}) \times (\text{Choices for 3rd digit}) \times (\text{Choices for 4th digit}) \]
\[ = 4 \times 5 \times 5 \times 5 \]

5. **Perform the multiplication**:
- First, calculate \(5 \times 5 \times 5 = 125\).
- Then, multiply by 4: \(4 \times 125 = 500\).

6. **Final Answer**: The total number of 4-digit numbers that can be formed is **500**.

### Summary:
The total number of 4-digit numbers that can be formed using the digits 0, 2, 3, 4, and 5 with repetition allowed is **500**.