Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The total number of different Pennsylvania license plate numbers that can be formed is \(26^3 \times 10^4.\)

Step 1: Calculating the possibilities for letters

Since the English alphabet consists of 26 letters and these letters can be repeated, there are 26 options for each of the three letter positions. Thus for the first part of the license plate, the total different possibilities is \[26 \times 26 \times 26 = 26^3.\]

Step 2: Calculating the possibilities for numbers

For the second part of the license plate that is composed of four numbers, 10 different numbers from 0 to 9 can be chosen for each position. Therefore, as with the letters, the possibilities for the number part turns out to be \[10 \times 10 \times 10 \times 10 = 10^4.\]

Step 3: Determining the total number of distinct license plates

As the license plate consists of two parts, the letters and the numbers, and each part is independent from each other, the total number of distinct license plates possible is the product of the possibilities calculated in the previous steps. Hence, it is \[26^3 \times 10^4.\] This is because the 'choice' of letters doesn't affect the 'choice' of numbers.

Key Concepts

Permutations

Understanding permutations is essential when it comes to counting the number of ways in which a set of things can be ordered or arranged. In our daily lives, we constantly arrange items, whether it be books on a shelf or apps on a phone screen. Permutations allow us to calculate the total possible arrangements when order matters.

In the case of Pennsylvania license plates, we're interested in the arrangement of letters and numbers. For the letter arrangement, imagine choosing a first, second, and third letter slot. The first slot can be filled by any of 26 letters, and since repetition is allowed, the second slot also has 26 options, as does the third. This scenario doesn't require all different letters; 'AAA' is just as valid as 'ABC'. Mathematicians would say there are 26 permutations of one letter taken three times.

If repetition was not allowed, the number of permutations would decrease because once a letter is used, it couldn't be chosen again. However, since license plate letters can repeat, the number of permutations is precisely 26 raised to the power of 3, which represents all the ways to arrange 3 letters when each can be any letter from the alphabet.

Exponential Notation

Exponential notation is a concise way to express repeated multiplication of the same number. It's a valuable tool for dealing with large numbers, such as when counting permutations, without writing out extensive multiplication. The exponential notation consists of a base number and an exponent. The base is the number being multiplied, and the exponent tells us how many times to use the base in multiplication.

In the Pennsylvania license plates example, we see the exponential notation in two instances: the three letters can be represented using 26 raised to the 3rd power, or \(26^3\), and the four digits can be represented as 10 raised to the 4th power, or \(10^4\). This notation tells us that we multiply 26 by itself three times for the letters and 10 by itself four times for the numbers. The exponent shows us how many slots we have to fill, and the base tells us how many choices there are for each slot.

Fundamental Counting Principle

The fundamental counting principle is a rule that allows us to count the number of ways multiple independent events can occur. It states that if one event can occur in 'm' ways and a second independent event can occur in 'n' ways, then the total number of ways both events can occur is the product of 'm' and 'n'.

Applied to our license plate problem, we have two independent events: selecting the three letters and the four digits. Since these events don't interfere with each other, we use the fundamental counting principle to find the total number of distinct license plates by multiplying the number of ways to arrange the letters (\(26^3\)) by the number of ways to arrange the numbers (\(10^4\)). This principle greatly simplifies our problem because it converts a complex counting issue into a simple multiplication of possibilities. By utilizing this principle along with the concept of permutations and exponential notation, we can efficiently determine the total number of potential Pennsylvania license plates.