Final answer:
To determine the maximum number of trials to guess the correct ATM PIN with the given conditions, one must consider all valid digit combinations that adhere to the constraints, including that all digits are different, the greatest digit is 7, and the sum of the pairs of digits are equal. The correct option is not mention.
Step-by-step explanation:
The question posed requires us to find the maximum number of trials necessary to guess a 4-digit ATM PIN code, given that all digits are different, the greatest digit is 7, and the sum of the first two digits equals the sum of the last two digits. To solve this, we need to consider possible combinations of digits ensuring that the aforementioned conditions are met.
To begin, we recognize that the greatest digit is 7, so our digits are constrained to 0-7. We also know that the first two and last two digits need to sum to the same total. With these conditions in place, we focus on organizing possible pairs of digits that could make up the first half and the second half of the PIN code.
We avoid duplication and combinations that do not meet the sum condition.
Assuming the largest digit 7 is part of the PIN, we can have the following pair constructions: (7,0), (6,1), (5,2), and (4,3). Each of these pairs can be arranged in 2! (factorial) ways, leading to 2 possible combinations per pair. Considering the conditions, the number of trials will be 4 pairs × 2 arrangements per pair × the number of ways to split the four chosen digits into two pairs.
After calculating the number of valid combinations that satisfy the sum condition, we determine the number of trials necessary to find the correct PIN code without repetition. The correct option is not mention.