Final answer:
The maximum number of trials required to open all locks when there are n locks and n keys is represented by the sum formula n(n+1)/2. Given the maximum is 105, solving n(n+1)/2 = 105 reveals that n = 14. So the correct answer is option B.
Step-by-step explanation:
When considering the question of how many trials are required to open all locks when there are n locks and n keys, the problem resembles a worst-case scenario in which each key is tried with each lock until all locks are opened. In the worst-case scenario, the first lock would take n trials (trying every key), the second lock would take n-1 trials (since one key is already found), and so forth, until the last lock takes just 1 trial. Therefore, the total number of trials can be represented by the sum of the first n natural numbers, given by the formula n(n+1)/2.
If the maximum number of trials is 105, we solve for n using the equation n(n+1)/2 = 105. The factors of 210 (since 105 is half of 210) are 14 and 15, which suggests that n is 14 since it must be a natural number and part of the equation n(n+1). Hence, n = 14.