Problem 41 - A Graphical Approach to College Algebra

Question

Use the fundamental principle of counting or permutations to solve each problem. License Plates For many years, the state of California used 3 letters followed by 3 digits on its automobile license plates. (a) How many different license plates are possible with this arrangement? (b) When the state ran out of new plates, the order was reversed to 3 digits followed by 3 letters. How many additional plates were then possible? (c) Several years ago, the plates described in part (b) were also used up. The state then issued plates with 1 letter, followed by 3 digits, and then 3 letters. How many plates does this scheme provide?

Step-by-Step Solution

Answer

(a) 17,576,000 (b) 17,576,000 (c) 456,976,000

Step 1: Calculate Plates with 3 Letters and 3 Digits

To find the total number of plates from the original 3-letter followed by 3-digit arrangement, we calculate the possibilities for letters and digits separately. Each position for a letter can have any of the 26 letters of the alphabet, and each digit can have any of the 10 numbers from 0 to 9. Therefore, the total number of combinations is calculated as follows:\[(26 \times 26 \times 26) \times (10 \times 10 \times 10) = 26^3 \times 10^3\]Calculating, we get:\[26^3 = 17576 \, \text{and} \, 10^3 = 1000\]Thus, the total number of different license plates is:\[17576 \times 1000 = 17,576,000\]

Step 2: Calculate Plates with 3 Digits and 3 Letters

For the reversed order (3 digits followed by 3 letters), the calculation remains similar, just the positions are swapped. Each position for a digit can be any of the 10 numbers, and each position for a letter can be any of the 26 letters. Thus, the number of combinations is:\[(10 \times 10 \times 10) \times (26 \times 26 \times 26) = 10^3 \times 26^3\]Using the previous calculations, the total remains:\[1000 \times 17576 = 17,576,000\]

Step 3: Calculate Plates with 1 Letter, 3 Digits, and 3 Letters

For the new arrangement of 1 letter, 3 digits, and 3 letters, each position has specific choices:- 1st position: 26 choices (1 letter)- Next 3 positions (digits): Each has 10 choices- Last 3 positions (letters): Each has 26 choicesThe total combinations can be calculated as:\[26 \times (10 \times 10 \times 10) \times (26 \times 26 \times 26) = 26 \times 10^3 \times 26^3\]Calculating this gives:\[26 \times 1000 \times 17576 = 456,976,000\]

Key Concepts

Permutations

Permutations are all about arranging items in an orderly manner. When the sequence of these items matters, we talk about permutations. This is crucial in various scenarios like permutations of numbers, letters, and even colors. For example, if you have three different letters and you want to see how many different ways you can arrange them, you're finding permutations of these letters. Here’s why this is important: arranging 'ABC' is different from 'CAB'.
In our license plate example, permutations help us figure out how many different ways we can arrange letters and numbers. Each different order counts as a new permutation. For instance, license plate 'ABC123' is different from 'CBA321'. Through permutations, we can systematically calculate these possibilities by looking at each position that needs to be filled.

Combinatorics

Combinatorics deals with counting, arranging, and combinations. It’s the mathematics of counting and helps us solve problems where we need to figure out how many ways we can pick and arrange certain items. This area of math is used in everything from planning business strategies to determining outcomes in board games.
The Fundamental Principle of Counting is a cornerstone of combinatorics, especially when determining the number of possible outcomes. It simply states that if you have multiple tasks, where each task has a specific number of choices, the total number of ways to perform these tasks is the product of the number of choices for each task. For instance, determining how many license plates can be created with specific arrangements of numbers and letters relies heavily on basic principles of combinatorics to calculate the total options effectively.

License Plate Combinations

Creating license plates is a practical application of permutations and combinatorics. Let's break down the example of license plate combinations. Initially, California used plates with three letters followed by three digits. Each letter has 26 possibilities and each digit has 10, leading us to calculate the total using the formula for combinations as \(26^3 \times 10^3\).
When the format changed due to running out on combinations, switching to three digits followed by three letters, the calculation remained fundamentally the same. Later, they switched to a format with one letter, three digits, and then three more letters, which increased complexity and the number of available plates drastically, showing the versatility and necessity of combinatorial principles in practical applications. These examples aptly illustrate how meticulous counting and arrangement are key in managing large sets of data like license plate numbers.