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The number lock of a suitcase has $4$ wheels, each labelled with ten 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?
$\frac{1}{5040}$
$\frac{1}{5040}$
$\frac{1}{5040}$
$\frac{1}{5040}$
Solution
The number lock has $4$ wheels, each labelled with ten digits i.e., from $0$ to $9 .$
Number of ways of selecting $4$ different digits out of $10$ digits $=^{10} C_{4}$
Now, each combination of $4$ different digits can be arranged in $\lfloor 4$ ways.
$\therefore$ Number of four digits with no repetitions $=^{10} C_{4} \times\left\lfloor 4=\frac{\lfloor {10}}{\lfloor {4\lfloor 6}} \times\lfloor 4=7 \times 8 \times 9 \times 10=5040\right.$
There is only one number that can be open the suitcase.
Thus, the required probability is $\frac{1}{5040}$.
