Final answer:
There are 2304 possible even four-digit numbers greater than 2000 that can be made without repeating any digits, calculated by considering the available options for each digit place and ensuring the last digit yields an even number.
Step-by-step explanation:
The question asks for the number of even four-digit numbers greater than 2000 that can be made without repeating any digit. First, let's consider the thousands place. Since the number must be greater than 2000, the options for the first digit are 2, 3, 4, 5, 6, 7, 8, and 9, giving us 8 possibilities. For the hundreds place, we now have 9 options remaining (0 and the numbers 1-9 excluding the digit from the thousands place). For the tens place, we have 8 options (those digits that haven't been used in the thousands or hundreds place). Finally, since the number must be even, the options for the ones place are 0, 2, 4, 6, and 8, but we've already used two digits, and cannot use 0 if it was used in the hundreds place, so we may only have 4 options left.
To find the total number of combinations, we calculate 8 options (for the first digit) × 9 options (for the second digit) × 8 options (for the third digit) × 4 options (for the fourth digit) = 2304 possible numbers. Therefore, the correct answer to how many even four-digit numbers greater than 2000 can be made if repetitions are not allowed is 2304, which is not one of the options provided in the question.