How many 4-digit numbers are there, when a digit may be repeated any number of times?

Explanation

To solve the problem of how many 4-digit numbers can be formed when digits may be repeated any number of times, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding 4-Digit Numbers**:
A 4-digit number must start with a non-zero digit (from 1 to 9) to ensure it is indeed a 4-digit number. The remaining three digits can be any digit from 0 to 9.

2. **Choosing the First Digit**:
The first digit can be any digit from 1 to 9. Therefore, there are 9 possible choices for the first digit.

3. **Choosing the Remaining Digits**:
The second, third, and fourth digits can each be any digit from 0 to 9. This gives us 10 choices for each of these digits.

4. **Calculating the Total Combinations**:
To find the total number of 4-digit numbers, we multiply the number of choices for each digit together:
- Choices for the first digit: 9
- Choices for the second digit: 10
- Choices for the third digit: 10
- Choices for the fourth digit: 10

Therefore, the total number of 4-digit numbers is:
\[ 9 \times 10 \times 10 \times 10 = 9 \times 1000 = 9000 \]

### Final Answer:
The total number of 4-digit numbers where a digit may be repeated any number of times is **9000**.

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