Problem 14 - Algebra for College Students

Question

A state has agreed to have its automobile license plates consist of two letters followed by four digits. State officials do not want to repeat any letters or digits in any license numbers. How many different license plates will be available? \(\quad 3,276,000\)

Step-by-Step Solution

Answer

The state can create 3,276,000 different license plates.

Step 1: Choose First Letter

The English alphabet has 26 letters. For the first position on the license plate (a letter), any of the 26 letters can be chosen. Thus, there are 26 possibilities for the first letter.

Step 2: Choose Second Letter

Since no letter can repeat, for the second position (another letter), we have 25 choices (one less, as the first choice has already been used). Thus, there are 25 possibilities for the second letter.

Step 3: Choose First Digit

A license plate allows for four digits from 0 to 9, totaling 10 available digits. Any of these 10 digits can be chosen for the third position (the first digit). Thus, there are 10 possibilities for the first digit.

Step 4: Choose Second Digit

Like the letters, no digits can repeat. Therefore, for the fourth position (the second digit), we have 9 choices (since one digit has already been used). This gives us 9 possibilities for the second digit.

Step 5: Choose Third Digit

For the fifth position (the third digit), we again have one fewer option, so there are 8 choices left (since two digits have already been used). Thus, there are 8 possibilities for the third digit.

Step 6: Choose Fourth Digit

For the sixth position (the fourth digit), we have 7 remaining choices (since three digits have already been used). Thus, there are 7 possibilities for the fourth digit.

Step 7: Calculate Total Combinations

Multiply the number of possible choices for each position to find the total number of possible license plates: \[ 26 \text{ (letters)} \times 25 \text{ (letters)} \times 10 \text{ (digits)} \times 9 \text{ (digits)} \times 8 \text{ (digits)} \times 7 \text{ (digits)} = 3,276,000. \]

Key Concepts

counting principle

To tackle problems involving permutations and combinations, the counting principle provides a fundamental approach. It helps us determine the total number of possible outcomes in a sequence of events. Think of it as a way to count every possible scenario without actually listing them all. When considering the counting principle, you multiply the number of choices for each event to get the total number of combinations. In the license plate problem, we segment the task into two parts: choosing letters and choosing numbers.

• Letters: For the first letter, there are 26 possible letters. For the second, there are 25 remaining letters after the first has been used. Thus, total options for the letters are 26 times 25.
• Digits: For the numbers, there are initially 10 possible digits (0 through 9). Subsequent digits reduce the available options by one each time, leading to calculations of 10, 9, 8, and 7 for each successive position.

By multiplying the number of possibilities for each element, we obtain the total number of unique license plates. This use of the counting principle ensures that every path of choices is considered, culminating in a complete solution.

factorials

Factorials, often denoted by an exclamation point (!), represent the product of all positive integers up to a certain number. They are fundamental in permutations and combinations due to their role in arranging items.The concept of factorials is closely related but is not directly used in every problem like our license plate example, where the counting principle suffices. However, consider this example:

• Definition: The factorial of a number, say 4!, is calculated as \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
• Application: Factorials help in arranging a set of items. For instance, arranging 5 different letters has \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \) unique permutations.

Understanding factorials enhances problem-solving abilities because it allows for quick calculations of possible arrangements when limitations like repetition are absent. In the license plate problem, although factorials weren't directly used, they are a vital tool in understanding how unique sequences are formed.

problem-solving

Problem-solving in mathematics often involves breaking down complex scenarios into manageable steps. With permutations and combinations, this often means understanding the constraints and utilizing basic principles effectively. For the license plate exercise, problem-solving begins by isolating the steps into choosing letters first, then digits. This approach simplifies the larger problem into smaller, more digestible parts:

• Identify the elements: letters and numbers.
• Apply constraints: no repetition of letters or digits.
• Count choices: sequentially work through each position.
• Calculate the result: use multiplication to find the total.

The process is streamlined using the counting principle, showcasing how methodical approaches lead to accurate outcomes. Good problem-solving involves not just knowing the methods, but also recognizing when and how to apply them, adapting to different problems with ease and confidence.