Question

The number lock of a suitcase has four wheels, each labelled with 10-digits, i.e., from 0 to 9. The lock opens with a sequence of four digits with no repeats. What is the probability of a person getting the right sequence to open the suitcase?
$
  {\text{A}}{\text{. }}\dfrac{{\text{1}}}{{{\text{5040}}}} \\
  {\text{B}}{\text{. }}\dfrac{{\text{3}}}{{{\text{5040}}}} \\
  {\text{C}}{\text{. }}\dfrac{{\text{7}}}{{{\text{5040}}}} \\
  {\text{D}}{\text{. None Of These}} \\
 $

Answer 1

Hint: In the given question, it is given that the number lock of a suitcase has four wheels, each labelled with 10 digits from 0 to 9. We find the number of the arrangement of these numbers in the four wheels keeping in mind that the repetition is not allowed. This number of arrangements gives the total four-digit number we can form without repetition from the given 10 digits that are, from 0 to 9. As only one sequence out of the total arrangement is correct, so, we can easily find the probability of getting the sequence right.

Complete step-by-step solution:
There are four wheels in a lock system. Out of the four wheels, the first wheel can have any one of the ten digits from 0 to 9 since repetition is not allowed. So, the second wheel can have any of the remaining 9 digits.
Similarly, the third wheel can have any of the remaining 8 digits.
And the fourth wheel can have any of the remaining 7 digits.
So, the number of the four digits lock code that can be formed without repetition of digits is given as:
$
  N = 10 \times 9 \times 8 \times 7 \\
   = 5040 \\
 $
So, the total number of four digits number formed $ = 5040$
But since the lock can open with only one of all the four-digit numbers, the required probability is $P = \dfrac{1}{{5040}}$.
Hence, the probability of a person getting the right sequence to open the suitcase is $\dfrac{1}{{5040}}$.

Hence the correct answer is option A.

Note: Some students might assume the total number of arrangements to be$10 \times 10 \times 10 \times 10$, and fail to get the correct answer. But it is very important to note that there is no repetition of numbers in the wheels so that the four wheels have four different numbers. So the total number of arrangements comes out to be $ = 10 \times 9 \times 8 \times 7 = 5040$, which gives the correct answer.