Question

A combination lock on a suitcase has 3 wheels each labelled with nine digits from 1 to 9. If an opening combination is a particular sequence of three digits with no repeats, what is the probability of a person guessing the right combination?
$\left( a \right)\dfrac{7}{{275}}$
$\left( b \right)\dfrac{1}{{501}}$
$\left( c \right)\dfrac{1}{{504}}$
$\left( d \right)$ None of these

Answer 1

Hint: In this particular question use the concept that there are 3 wheels in a combination lock and no digit is repeated so the first digit is filled by 9 ways as the number of digits available is from 1 to 9, then use the concept that when first digit is filled so one digit is exhausted so remaining digits are 8 so second digit is filled by 8 ways, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
A three wheel combination lock.
Every wheel has digits from 1 to 9.
Now we have to find the probability of a person guessing the right combination if no digit is repeated.
As we know that the probability is the ratio of favorable number of outcomes to the total number of outcomes.
$ \Rightarrow P = \dfrac{{{\text{favorable number of outcomes}}}}{{{\text{total number of outcomes}}}}$
Know as we know that there is only 1 right three digit combination, so the favorable number of outcomes = 1.
Now find the total number of three digit combinations.
As there are 9 digits available so the first digit of a three digit combination is filled by 9 ways.
Now as one digit is exhausted and no digit is repeated so the number of ways to fill the second digit of a three digit combination is 8.
Similarly the number of ways to fill the third digit of a three digit combination is 7.
So the total number of three digit combinations is the multiplication of the above cases.
So the total number of three digit combinations = $9 \times 8 \times 7 = 504$
So the total number of outcomes is 504.
So the probability is, $P = \dfrac{1}{{504}}$.
So this is the required answer.
Hence option (c) is the correct answer.

Note: Whenever we face such types of questions the key concept we have to remember is the definition of probability which is stated above in the middle of the solution, so first find out the favorable as well as total number of outcomes as above, then divide them so we will get the required probability.