Question

A letter lock contains 3 rings, each ring containing 5 different letters. Determine the maximum number of false trials that can be made before the lock is opened.

Answer 1

Hint: Find the total number of attempts and subtract 1 from the total because that would be the successful one. We have 5 letters and 3 rings, so we can arrange the letters in each of these 3 rings in \[5\times 5\times 5\] . This gives us the total possible attempts.

Complete step by step answer:
Letter lock is a lock which is not opened by any keys, but it can be opened by the password or we can say by arranging all the letters in a row perfectly, that can unlock it. This password can be of any length depending upon the manufacturer.
Here, we are given a letter lock which contains 3 rings, each ring containing 5 letters.
That means the password is a row of 3 letters.
We can see the situation clearly as:
\[\begin{align}
  & -- \\
 & 5\text{ 5 5} \\
\end{align}\]
Three blanks are there and in each blank 5 letters can be put one by one.
Since, there are 5 letters in each ring, the maximum number of possible attempts to unlock the lock would be \[5\times 5\times 5=125\]
The letter in the rings can be arranged in a total 125 ways.
Since, 125 arrangements are total arrangements, it must contain all the unsuccessful as well as one successful attempt.
Question here asks for the attempt before the lock is opened, that means, we are asked to give the number of all the unsuccessful attempts.
Therefore, all unsuccessful attempts,
\[\begin{align}
  & \Rightarrow \text{Total attempts - 1} \\
 & \Rightarrow \text{125-1} \\
 & \Rightarrow \text{124} \\
\end{align}\]
So, the required attempts or number of false trials is 124.

Note: Students before observing the problem clearly can make mistakes by giving the answer as 125. It is necessary to understand the question thoroughly. Some students commit the mistake of taking \[5\times 3\] while finding the total attempts possible. But, it is completely wrong, here we have 3 rings and 5 letters in each to try out, so 5 must be multiplied with itself 3 number of times.