The digits 2, 5, 0, 6 and 8 are used to form five-digit numbers without repetition. What is the difference between the greatest and the least such five-digit numbers?

Introduction / Context:
This problem is about forming numbers from given digits under specific rules and then comparing those numbers. You are given the digits 2, 5, 0, 6 and 8 and asked to form the greatest and least possible five digit numbers without repeating any digit. After that, you must compute the difference between these two extreme values. This tests place value understanding and logical arrangement of digits rather than heavy computation.

Given Data / Assumptions:

- Available digits are 2, 5, 0, 6 and 8. - Each digit can be used at most once in a number. - We must form valid five digit numbers, so the leading digit cannot be 0. - We need the greatest and least such numbers and their difference.

Concept / Approach:
To form the greatest five digit number, we place the largest available digit in the highest place value (ten thousand place), then the next largest in the thousand place and so on, making sure we do not put 0 in the first position. To get the least five digit number, we place the smallest non zero digit in the first place, then arrange the remaining digits in ascending order from left to right, including 0 after the first digit. Once we have both numbers, we subtract the smaller from the larger to find the required difference.

Step-by-Step Solution:
Step 1: List the digits in increasing order: 0, 2, 5, 6, 8. Step 2: To form the greatest five digit number, choose the largest digit for the first place. That is 8. Step 3: For the remaining places, use the remaining digits in descending order: 6, 5, 2 and 0. Step 4: The greatest five digit number is therefore 86520. Step 5: To form the least five digit number, the first digit cannot be 0, so choose the smallest non zero digit, which is 2, for the ten thousand place. Step 6: Arrange the remaining digits 0, 5, 6 and 8 in ascending order for the remaining places: 0, 5, 6, 8. Step 7: The least five digit number is 20568. Step 8: Now compute the difference: 86520 - 20568. Step 9: Subtract 20568 from 86520. First subtract 20000 to get 66520, then subtract 568 to get 65952. Step 10: Thus, the required difference is 65952.

Verification / Alternative check:
You can quickly verify the subtraction using column method. Writing 86520 minus 20568 and borrowing as needed will yield the same result of 65952. Also, check that no other arrangement can produce a larger or smaller five digit number under the conditions. Any attempt to place a digit smaller than 8 in the first position of the greatest number or a digit larger than 2 in the first position of the least number will make the numbers less extreme, so the chosen arrangements are indeed optimal.

Why Other Options Are Wrong:
Values like 69552, 65925 and 63952 represent subtracting from slightly different largest or smallest numbers, often due to misplacing 0 or mixing up the digit order. For example, using 85620 as the greatest or 25068 as the least will alter the difference. Only 65952 corresponds to correctly formed extreme numbers that respect place value and the no repetition rule.

Common Pitfalls:
A typical mistake is putting 0 in the ten thousand place when forming the smallest number, which would create a four digit number rather than a valid five digit one. Another error is not strictly sorting digits for the largest or smallest arrangements, especially when 0 is involved. Remember that the leading digit of a multi digit number can never be 0, and always order remaining digits carefully once the first position is fixed.

Final Answer:
The difference between the greatest and least such five digit numbers is 65952, which corresponds to option C.