Problem
$-5 \cdot$ (3 marks) A lock has 5 dials, each with the numbers 0 through 9 as seen in the picture on the right. To unlock the lock, the correct sequence of numbers must be dialed. However, the lock is faulty, and it is considered unlocked if the correct sequence is dialed in any order of the dials. It is known that the correct password consists of two different pairs of the same numbers and one different number from the rest eg. 21182. What is the probability of unlocking the lock by randomly dialing the numbers?
Solution
Step 1 ：Find the total number of possible sequences: \(10^5 = 100000\)
Step 2 ：Find the number of unlocking sequences: \(\binom{10}{2} \cdot \binom{8}{1} \cdot \frac{5!}{2! \cdot 2!} = 10800\)
Step 3 ：Calculate the probability: \(\frac{10800}{100000} = 0.108\)
Step 4 ：\(\boxed{0.108}\)
