Problem:
A letter lock consists of three rings each marked with 10 different letters. Find the number of ways in which it is possible to make an unsuccessful attempt to open the lock.

Solution:
To solve this problem, we need to consider the possible number of unsuccessful attempts to open the letter lock.

Understanding the Letter Lock:
The letter lock consists of three rings, and each ring is marked with 10 different letters. Therefore, each ring has 10 possible positions to be set at a time.

Calculating the Number of Successful Attempts:
To calculate the number of successful attempts to open the lock, we need to consider that for each ring, there is only one correct position. Therefore, the total number of successful attempts is 1.

Calculating the Number of Unsuccessful Attempts:
To find the number of unsuccessful attempts, we need to consider all the possibilities where the lock is not opened successfully.

Approach:
To calculate the number of unsuccessful attempts, we need to count all the possible combinations of the lock positions, excluding the correct position. We can calculate this by subtracting the number of successful attempts from the total number of possible combinations.

Calculating the Total Number of Possible Combinations:
Since each ring has 10 different letters, there are 10 possible positions for each ring. Therefore, the total number of possible combinations is given by:
Total number of combinations = (Number of positions in ring 1) x (Number of positions in ring 2) x (Number of positions in ring 3)
= 10 x 10 x 10
= 1000

Calculating the Number of Unsuccessful Attempts:
To calculate the number of unsuccessful attempts, we subtract the number of successful attempts from the total number of possible combinations:
Number of unsuccessful attempts = Total number of possible combinations - Number of successful attempts
= 1000 - 1
= 999

Therefore, the number of ways in which it is possible to make an unsuccessful attempt to open the lock is 999.