A letter lock contains three rings each marked with fifteen different letters, find in how many ways it is possible to make an unsuccessful attempt to open the lock.

Hint: To solve the question, we have to calculate the total number of attempts, by applying the formula which states, if a lock contains r rings each marked with n different values, then the number of attempts to open the lock is equal to \[{{n}^{r}}\] .Thus, we can calculate the number of ways of it is possible make an unsuccessful attempt by subtracting the number of successful attempts from the total number of attempts.

Complete step-by-step answer:
We know that, if a lock contains r rings each marked with n different values, then the number of attempts to open the lock is equal to \[{{n}^{r}}\]
Thus, by applying the above given formula for the letter lock contains three rings each marked with fifteen different letters, we get
The number of attempts is equal to \[{{15}^{3}}=15\times 15\times 15=3375\]
We know that only one key can open the letter lock, this implies that the number of successful attempts = 1
The number of unsuccessful attempts = The total number of attempts - the number of successful attempts
= 3375 - 1
= 3374
Thus, the number of ways it is possible to make an unsuccessful attempt to open the lock is equal to 3374.

Note: The possibility of mistake can be not applying the formula for calculating the total number of attempts, which states, if a lock contains r rings each marked with n different values, then the number of attempts to open the lock is equal to \[{{n}^{r}}\] .The other possibility of mistake can be not analysing that only one key can open the letter lock, which implies that the number of successful attempts is equal to 1.