Question

A letter lock contains 5 rings each marked with four different letters. The number of all possible unsuccessful attempts to open the lock is

• A. 625
• B. 1024
• C. 624
• D. 1023

Hint:

For "unsuccessful attempts" subtract the successful one from total outcomes.

Answer 1

The Correct Option is D

To solve the problem of determining the number of possible unsuccessful attempts to open the lock, we first need to understand the mechanism of the letter lock described.

This letter lock consists of 5 rings, each marked with 4 different letters. The task is to find the number of unsuccessful attempts to open the lock using different combinations of these letters.

Step-by-Step Explanation:

1. Total Possible Combinations:
   Each ring has 4 different letters. Since there are 5 rings, and each ring is independent of the others, the total number of possible combinations is calculated by multiplying the number of options per ring, as follows: \(4^5\).
2. Calculate:
   Let's compute the total possible combinations: \(4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 1024\).
3. Successful Combination:
   Among these 1024 combinations, there is one correct combination that unlocks the lock.
4. Unsuccessful Attempts:
   To find the number of unsuccessful attempts, we need to exclude the successful combination from the total possible ones: \(1024 - 1 = 1023\).

Conclusion:

The number of all possible unsuccessful attempts to open the lock is 1023.

Thus, the correct answer is: 1023.