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Combinatorics is a fascinating field of mathematics that deals with counting, arrangement, and combination of objects. Imagine you have a basket of fruits, each with its own unique type. Combinatorics helps us understand how many different ways we can select fruits from this basket. In the context of our license plate problem, we utilize combinatorics to determine how many different possible outcomes exist when creating a license plate. For each slot on the plate, we have a set of options from which to choose. For instance, each of the three digit positions can have any one of 10 digits, and each of the three letter positions can have any one of 26 letters.By applying the basic principle of multiplication, combinatorics allows us to calculate the total number of outcomes by multiplying the number of options for each slot. Therefore, for three digits followed by three letters, the calculation becomes a multiplication of all possible combinations for digits with all possible combinations for letters, i.e., \(10 \times 10 \times 10\) for the digits and \(26 \times 26 \times 26\) for the letters.

Random selection is a key concept in probability, implying that each item or event from a set is equally likely to be chosen. Think about picking a card from a well-shuffled deck; each card has an equal chance of being drawn. In our license plate problem, we assume that each possible license plate number has an equal chance of being selected. This means if you pick a license plate randomly, each one of the 17,576,000 potential combinations is equally likely. Understanding random selection is critical when calculating probabilities because it ensures fairness and uniformity. For instance, if your license plate is randomly chosen from all possible plates, the likelihood of your specific plate being selected is simply one divided by the total number of plates, given the randomness of the process.

License plate combinations are created using a systematic process of selecting characters from specified sets. In many places, license plates have a standard structure; ours has three numbers followed by three letters.To figure out how many possible license plates there are, you need to understand how these characters are combined.

• The first part is the creation of digit combinations: With each of the three positions capable of having any digit from 0 to 9, we calculate this as \(10 \times 10 \times 10 = 1,000\).
• Next, we handle the letter combinations: With each of the three positions having possible selections from A to Z, we compute this as \(26 \times 26 \times 26 = 17,576\).
• Finally, by multiplying the two results, we get the total number of license plates: \(1,000 \times 17,576 = 17,576,000\) unique plates.

Understanding these combinations helps clarify how probabilities are assessed, by showing how each individual choice contributes to the total possibilities.