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

Answer:

To find the number of 4-digit numbers when a digit may be repeated any number of times, we can use the concept of permutations and combinations.

Explanation:

When a digit may be repeated any number of times, it means that each digit in the 4-digit number can have any value from 0 to 9.

Approach:
To find the number of 4-digit numbers, we need to consider the following:
- The first digit cannot be zero (as it will no longer be a 4-digit number).
- Each digit in the 4-digit number can have any value from 0 to 9.

Step 1:
Count the number of possibilities for each digit from 1 to 9 (excluding zero). Since each digit can have any value from 0 to 9, there are 10 possibilities for each digit.

Step 2:
Count the number of possibilities for the first digit. Since the first digit cannot be zero, there are 9 possibilities for the first digit.

Step 3:
Count the number of possibilities for the remaining 3 digits. Since each digit can have any value from 0 to 9, there are 10 possibilities for each of the remaining 3 digits.

Step 4:
Multiply the number of possibilities for each step. The total number of 4-digit numbers is calculated as follows:

Number of 4-digit numbers = Number of possibilities for the first digit * Number of possibilities for the remaining 3 digits

Number of 4-digit numbers = 9 * 10 * 10 * 10 = 9000

Therefore, the correct answer is option 'B' - 9103.