Question

The number of unsuccessful attempts that can be made by a thief to open a number lock having 3 rings in which each contains 6 numbers is
A) 205
B) 200
C) 210
D) 215

Answer 1

Hint: In this question, we have given 3 rings and each ring contains 6 numbers. A thief wants to open the lock and we have to calculate the number of unsuccessful attempts made by the thief. As number locks can be opened by only one number, so the number of successful attempts is 1. So, the number of unsuccessful attempts is equal to total attempts – 1.

Complete step by step solution: 1) We have three rings and each ring contains 6 numbers. So, the total number of successful attempts are as shown in the diagram.

2) Here you can see that the first ring can be filled in 6 ways as it contains 6 letters, second rings also have 6 numbers so it can be filled in 6 ways and similarly the third ring can be filled in 6 ways. So, the total number of attempts made for the lock to open is $6 \times 6 \times 6$ i.e.
$ \Rightarrow {A_T} = 6 \times 6 \times 6$, where ${A_T}$ is total attempts to open the lock.
3) To open a number lock there is only one way because every lock has unique code and always opens with one number. So, the number of successful attempts is 1. i.e.
$ \Rightarrow {A_S} = 1$, where ${A_S}$ is the number of successful attempts.
4) Number of unsuccessful attempts a thief can make to open a lock is equal to ${A_T} - {A_S}$ i.e.
$ \Rightarrow {A_{US}} = {A_T} - {A_s}$, where ${A_{us}}$ is the number of unsuccessful attempts made to open the lock.
$\begin{gathered}
   \Rightarrow {A_{US}} = {6^3} - 1 \\
   \Rightarrow {A_{US}} = 216 - 1 \\
   \Rightarrow {A_{US}} = 215 \\
\end{gathered} $

Hence option D is the correct answer.

Note: 1) Mostly during solving this question, students make mistakes and mistakes made by them is to take a number of attempts as $6 \times 5 \times 4 = 120$, which is wrong. As each ring has 6 digits on its own so we will select from 6 digits for each ring.
2) The second mistake done by them is to consider total attempts as the number of unsuccessful attempts but your answer gets wrong. As there is one successful attempt to open a lock as every lock has its unique code. Take care of these types of mistakes.
3) We have given 4 options and none of the options contains the possibility of making a mistake but if they included that option then your answer will be wrong.