91Ó°ÊÓ

Permutation and combination are fundamental mathematical concepts that deal with the arrangement of objects. To understand the difference, think of permutations as arrangements where the order matters, and combinations where the order doesn't.

For instance, let's consider a simpler version of the exercise problem. If we had to arrange two distinct digits, the number of permutations would be the product of the available options for each place. In a 2-digit number where both digits must be different, and let's say the first digit can be anything from 1 to 9, and the second digit any number except the first one chosen, the number of permutations is calculated by multiplying the possibilities for each. Since there are 9 choices for the first digit and 8 for the second, we have a total of 72 permutations. This principle is exactly what's applied in Step 2 of the provided exercise, building off from the given that the first of the three digits is always 5.

Factorial Notation

When dealing with permutations where all elements are different, we often use factorial notation, denoted as \(n!\), where \(n\) stands for the total number of objects to arrange. Factorial of any number \(n\) is the product of all positive integers less than or equal to \(n\). But in the context of the exercise, since the arrangement of the last two digits is constrained (digits must be different), factorial notation is not directly applied.

In probability theory, events are considered independent if the occurrence of one event does not affect the probability of another. This concept is essential in calculating probabilities for a series of events where each event has no influence on the others.

In the exercise, the choosing of each digit for the license plate is an independent event. Each digit is selected without any restriction from the previous choices, except for the condition that they must be different. Once we choose the first digit (already determined to be 5), the selection of the second digit is independent but has to exclude the first digit. Likewise, the third digit's selection is independent of the second digit's choice while ensuring that it is different from both the previous digits.

This independence simplifies our calculations, allowing us to simply multiply the number of choices for each digit to get the total possibilities, as seen in Step 2, resulting in 72 different possible license numbers.

Probability theory is a branch of mathematics concerned with analyzing random events. It provides the principles and methods to determine the likelihood of various possible outcomes. Often, it involves calculations based on permutations and combinations to determine the number of favorable outcomes to the total possible outcomes.

In our exercise, we aren't specifically asked to calculate the probability but rather to count the number of possible outcomes—72 valid registrations. However, understanding probability theory is intrinsic in recognizing why we multiply the independent choices for each digit together, as opposed to adding them or using another operation.

If we were to ask what is the probability that a random license plate matches the witness description, assuming all combinations of letters and numbers are equally likely, we would divide our number of favorable outcomes (72) by the total number of possible license plates given the constraints. Probability theory provides a systematic way to make informed statements about such random events, even when there's uncertainty involved, much like the situation at hand with the missing digits in the accident witness's report.