Problem 72 - Basic Statistics for Business and Economics with Student

Question

The state of Maryland has license plates with three numbers followed by three letters. How many different license plates are possible?

Step-by-Step Solution

Answer

17,576,000 different license plates are possible.

Step 1: Understand the problem

The problem involves determining the total number of possible license plate combinations in Maryland, which have a format consisting of three digits followed by three letters.

Step 2: Determine the possibilities for the numbers

A license plate has three numbers, and each number is a digit ranging from 0 to 9. Thus, there are 10 possible choices for each digit. Therefore, the total number of possibilities for the numbers is calculated as: \[ 10 \times 10 \times 10 = 10^3 = 1000 \].

Step 3: Determine the possibilities for the letters

The license plate also includes three letters, and each letter can be any one of the 26 letters in the English alphabet. Thus, there are 26 possible choices for each letter. Therefore, the total number of possibilities for the letters is calculated as: \[ 26 \times 26 \times 26 = 26^3 = 17576 \].

Step 4: Calculate the total number of license plates

To find the total number of different license plates possible, multiply the number of possibilities for the numbers by the number of possibilities for the letters: \[ 1000 \times 17576 = 17,576,000 \].

Step 5: Conclusion

The total number of different license plates possible in Maryland, based on the given format, is 17,576,000.

Key Concepts

Probability

Probability is a fundamental concept in combinatorics and mathematics. It essentially measures the chance or likelihood of a particular event occurring. In our license plate problem, we're analyzing combinations rather than probabilities. We calculate the total number of outcomes in constructing a license plate. However, understanding these combinations lays the groundwork for calculating probabilities in more complex situations.

Whenever you're working with probabilities, you'll encounter scenarios where you need to count possible outcomes or combinations. These combinations help define the total number of events that could occur. In the license plate example, if hypothetically we were interested in the probability of a specific license plate, we would compare that one unique combination against the total possible number, which in this exercise is 17,576,000.

To sum up, while this exercise primarily deals with counting combinations, it ties directly into the broader concept of probability. Understanding how to calculate potential combinations is a critical skill in determining probabilities in more advanced scenarios.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to determine whether conclusions drawn from ideas are valid. In our context of the license plate problem, mathematical reasoning aids in breaking down the problem into smaller, manageable parts.

Here, the problem involves calculating the number of distinct license plates, which requires reasoning through the structure and rules of the problem.

• First, recognize the structure: a sequence of three numbers followed by three letters.
• Second, determine the number of choices: each number can be 0-9, and each letter can be A-Z.
• Third, logically combine these parts to calculate the total: multiply the number of choices for numbers by the number of choices for letters.

Mathematical reasoning ensures each step is logically sound, leading to the correct conclusion of 17,576,000 possible combinations. This type of reasoning, combining logical structure with calculations, is fundamental in all kinds of mathematical problem-solving.

Problem-solving in Mathematics

Problem-solving in mathematics often involves applying a systematic approach to find solutions to complex situations. When solving the Maryland license plate problem, a step-by-step approach was essential.


This kind of problem-solving involves:

• Understanding the problem: Recognize that you need to find all possible combinations of numbers and letters in a given format.
• Breaking the problem into smaller parts: Separating the numbers from letters and calculating possible combinations for each.
• Applying the principles of combinatorics: Using permutations and combinations to count possibilities accurately.

By systematically following these steps, the problem becomes less daunting, and a solution is reached efficiently. Mastering problem-solving in mathematics helps tackle even more complex problems beyond this exercise, making it an essential skill for all math students.