Final answer:
There are a total of 4 numbers that can be created which are greater than 500 and do not repeat any digits using the set {2, 3, 4, 5}. The numbers are 532, 523, 452, and 453.
Step-by-step explanation:
The question asks for the creation of numbers greater than 500 without repeating digits using some or all digits from the set {2, 3, 4, 5}. Since the numbers must be greater than 500, the first digit must be either 5 or a number larger (the digits we have only include one number larger, which is 5).
There are two possibilities for the hundreds place: 5 or 4. If 5 is in the hundreds place, then we have two options for the tens place (3 or 2) and only one option for the units place, which is the remaining number that we didn't use in the tens place. This gives us 2 possible numbers:
If we place 4 in the hundreds place, we must place 5 in the tens place (as no other options are left to make the number greater than 500), and then we have two options for the units place (either 2 or 3). This gives us 2 more numbers:
Adding the possibilities up, we get a total of 4 numbers that can be created without repeating digits and being greater than 500 using the digits 2, 3, 4, and 5. Therefore, the answer to how many numbers were written is 4, which is not an option in the question suggesting there may have been an error in either the construction of the question or the provided answer choices.