Problem:
Find the number of even numbers that can be formed with the digits 0, 1, 2, 3, 4, where none of the digits are repeated in any number.

Solution:
To find the number of even numbers, we need to consider the following conditions:
1. The last digit must be even (0, 2, or 4).
2. The remaining digits can be arranged in any order.

Step 1: Choosing the last digit
Since the last digit must be even, we have 3 choices (0, 2, or 4).

Step 2: Choosing the remaining digits
The remaining digits can be arranged in any order. We have 5 digits to choose from (0, 1, 2, 3, 4), but we need to choose one less digit each time.

- For the first digit, we have 5 choices.
- For the second digit, we have 4 choices.
- For the third digit, we have 3 choices.
- For the fourth digit, we have 2 choices.
- For the fifth digit, we have 1 choice.

Therefore, the total number of arrangements for the remaining digits is 5 x 4 x 3 x 2 x 1 = 120.

Step 3: Calculating the total number of even numbers
To find the total number of even numbers, we multiply the choices from Step 1 and Step 2.

Total number of even numbers = 3 x 120 = 360.

Answer:
The number of even numbers that can be formed with the digits 0, 1, 2, 3, 4 (where none of the digits are repeated) is 360.