Question: A small combination lock on a suitcase has 3 wheels, each labeled with the 10 digits from 0 to 9…

A small combination lock on a suitcase has 3 wheels, each labeled with the 10 digits from 0 to 9. If an opening combination is a particular sequence of 3 digits with no repeats, what is the probability of a person guessing the right combination?

The probability of a person guessing the right combination is (Type a fraction. Simplify your answer.)

Solution

To determine the probability of guessing the right combination for the lock, we need to find the total number of possible combinations and the number of successful outcomes (i.e., guessing the right combination). Each wheel on the lock can be set to a digit from 0 to 9, with no repeats allowed in a 3-digit sequence. Therefore, we use permutations to calculate the total possible combinations. The number of permutations of 10 digits taken 3 at a time is given by the formula: \[ P(n, k) = \frac{n!}{(n-k)!} \] where \( n = 10 \) (the total number of digits) and \( k = 3 \) (the number of digits in the combination). Calculate the number of permutations: \[ P(10, 3) = \frac{10!}{(10-3)!} \] Calculate \( 10! \) and \( 7! \): \[ 10! = 10 \times 9 \times 8 \times 7! = 3628800 \] \[ 7! = 5040 \] Now, calculate the permutations: \[ P(10, 3) = \frac{3628800}{5040} = 720 \] Thus, there are 720 possible combinations. The probability of guessing the right combination is the ratio of successful outcomes (1 successful outcome, the right combination) to the total outcomes (720 possible combinations): \[ \text{Probability} = \frac{1}{720} \] Therefore, the probability of a person guessing the right combination is \( \frac{1}{720} \).