A letter lock consists of three rings, each ring marked with six different letters. What is the maximum number of distinct unsuccessful attempts that can be made to open the lock if the correct combination is unknown?

Introduction / Context:
This problem is about counting the number of different combinations on a simple letter lock. Each ring can show one of six letters, and the combination consists of the letters shown by the three rings together. We are asked to find the maximum number of incorrect attempts possible, which is simply all possible combinations minus the one correct combination.

Given Data / Assumptions:
The lock has 3 rings. Each ring has 6 different letters. Every ring can independently show any one of its 6 letters at a time. Exactly one combination is the correct one that opens the lock. All other combinations are unsuccessful attempts.

Concept / Approach:
The total number of possible combinations is found by multiplying the number of choices for each ring. Since there are 6 choices on each of the 3 rings, the total combinations equals 6 * 6 * 6 which is 6^3. If we assume that each distinct combination is tried at most once, the maximum number of unsuccessful attempts occurs when we try every combination except the correct one. Therefore the number of unsuccessful attempts is total combinations minus 1.

Step-by-Step Solution:
Step 1: Compute the number of possible letter combinations on the lock. Step 2: For ring 1, there are 6 letter choices. Step 3: For ring 2, there are also 6 letter choices. Step 4: For ring 3, there are again 6 letter choices. Step 5: Total combinations = 6 * 6 * 6 = 6^3 = 216. Step 6: Exactly one of these 216 combinations is the correct one. Step 7: Maximum unsuccessful attempts = total combinations - 1 = 216 - 1 = 215.

Verification / Alternative check:
An alternative way to view this is to imagine systematically listing all possible three letter sequences using the 6 letters on each ring. There are 6 possibilities for the first letter, 6 for the second, and 6 for the third, giving 216 sequences. If you tried every sequence exactly once, you would be guaranteed to use the correct one at some attempt. The number of wrong ones is simply one less than 216, which is 215, confirming the result.

Why Other Options Are Wrong:
The value 216 is the total number of possible combinations, including the correct one, so it is too large for the number of unsuccessful attempts. Values like 268 and 254 exceed the total number of combinations or do not have a logical basis here. The value 200 is smaller than the total combinations minus one and implies not all possible incorrect combinations are tried. Only 215 correctly equals 6^3 minus one.

Common Pitfalls:
Some students forget to subtract 1 and give 216 as the answer, misinterpreting the question as asking for total combinations rather than unsuccessful ones. Others may miscalculate 6^3 or confuse combinations with permutations, although in this case they coincide because the positions (rings) are distinct. Carefully reading that the question is about unsuccessful attempts and recognizing the need for total minus one avoids these issues.

Final Answer:
The maximum number of distinct unsuccessful attempts to open the lock is 215.