The sum of all four-digit even numbers formed using the digits 0, 1, 2, 3, 4, and 5 with repetition allowed.

To find the sum of all four-digit even numbers formed using the given digits, we need to consider the possible combinations and calculate the sum for each combination.

Step 1: Determine the possible combinations
Since repetition of digits is allowed, we have six choices for each digit in the four-digit number. Therefore, the total number of possible combinations is 6*6*6*6 = 6^4 = 1296.

Step 2: Identify the even numbers
Since we are looking for even numbers, the last digit must be either 0, 2, or 4. The remaining three digits can be any of the six given digits. Therefore, there are a total of 3*6*6*6 = 648 even four-digit numbers that can be formed.

Step 3: Calculate the sum
To calculate the sum of all these even four-digit numbers, we need to find the sum of each digit's place value for all the possible combinations.

Thousands place:
Since the thousands place can only be 1, 2, 3, 4, or 5, and there are 648 numbers in total, the sum of all the thousands places is (1+2+3+4+5) * 648 = 15 * 648 = 9720.

Hundreds place:
Since the hundreds place can be any of the six digits, the sum of all the hundreds places is (0+1+2+3+4+5) * 648 = 15 * 648 = 9720.

Tens place:
Similarly, the sum of all the tens places is (0+1+2+3+4+5) * 648 = 15 * 648 = 9720.

Units place:
Since the units place can only be 0, 2, or 4, the sum of all the units places is (0+2+4) * 648 = 6 * 648 = 3888.

Step 4: Calculate the final sum
To find the sum of all the four-digit even numbers, we add up the sums of each place value:

9720 + 9720 + 9720 + 3888 = 33048.

Therefore, the sum of all the four-digit even numbers formed using the digits 0, 1, 2, 3, 4, and 5 with repetition allowed is 33048.

To summarize:
- There are 1296 possible combinations of four-digit numbers using the digits 0, 1, 2, 3, 4, and 5 with repetition allowed.
- Out of these combinations, 648 are even numbers.
- The sum of the thousands place is 9720, as there are 648 numbers in total and the thousands place can be any of the digits 1, 2, 3, 4, or 5.
- The sum of the hundreds place is also 9720, as the hundreds place can be any of the six digits.
- The sum of the tens place is 9720,