Problem 21 - Precalculus with Limits

Question

In the state of Pennsylvania, each standard automobile license plate number consists of three letters followed by a four-digit number. How many distinct license plate numbers can be formed in Pennsylvania?

Step-by-Step Solution

Answer

The total number of distinct license plate numbers possible in Pennsylvania is \(26^3 * 10^4\)

Step 1: Determine the number of possibilities for the letters

Since there are 26 letters and the license plate uses 3, with repetition allowed, the calculation is \(26^3\). That is, there are 26 options for each of the 3 letter positions.

Step 2: Determine the number of possibilities for the digits

Next, consider the four digits. There are 10 options (0-9) for each digit, with four digits in total. Therefore, there are \(10^4\) options for the digits.

Step 3: Calculate the total number of distinct license plates

To find the total number of possible license plates, multiply the number of possibilities for the letters by the number of possibilities for the numbers, which is \(26^3 * 10^4\).

Key Concepts

Permutations

When we talk about permutations in combinatorics, we're exploring the different ways we can arrange a set of objects. In our license plate example, the arrangement of three letters can be seen as a permutation problem where repetition is allowed. That's because each letter slot can contain any letter from A to Z independently, creating a distinct sequence each time.

Imagine you have a box of letter tiles, each representing one letter of the alphabet. For the first spot on the license plate, you can pick any of the 26 letters. After picking the first, you still have 26 choices for the second spot because repetition is permitted. The same goes for the third spot. The total number of permutations is calculated by multiplying the number of choices at each step, which is where the counting principle comes into play.

Exponential Notation

Exponential notation is a convenient way to express repeated multiplication of the same number. It simplifies writing and calculating large numbers, which is especially handy in combinatorics. A number like \(26\) raised to the third power, denoted as \(26^3\), means that we multiply \(26\) by itself three times.

This compact form lets us quickly understand the scale of combinations without manually multiplying. For the license plate problem, we are dealing with \(26^3\) for letters, which represents 26 options for the first letter, multiplied by 26 for the second, and again by 26 for the third. Similarly, \(10^4\) for the four digits follows the same principle, indicating ten options for each of the four positions on the plate.

Counting Principle

The counting principle allows us to calculate the total number of combinations or arrangements in a scenario with multiple stages or parts. This fundamental rule of counting is a cornerstone in permutation problems. It states that if one event can occur in 'm' ways and a subsequent event can occur in 'n' ways, the total ways both can occur is 'm * n'.

In the context of our license plate task, we apply the principle twice. First, to find the permutations of the three-letter sequence (\(26^3\)), and second, for the four-digit sequence (\(10^4\)). To get the total distinct license plate numbers, we multiply these two results together, as they represent independent events. Here's where combinatorics beautifully simplifies complex tasks into more manageable calculations.