Problem 23 - Algebra and Trigonometry

Question

License Plates Standard automobile license plates in California display a nonzero digit, followed by three letters, followed by three digits. How many different standard plates are possible in this system?

Step-by-Step Solution

Answer

There are 158,184,000 possible standard license plates in this system.

Step 1: Calculate Options for Non-Zero Digit

Consider the first digit on the license plate, which cannot be zero. Therefore, it can range from 1 to 9. There are 9 possible choices for this non-zero digit.

Step 2: Calculate Options for Three Letters

Each of the next three characters must be a letter. Given that there are 26 letters in the English alphabet, each of the three positions can have 26 different choices. For three letters, the total number of combinations is calculated as \(26 \times 26 \times 26\).

Step 3: Calculate Options for Three Digits

The final three positions on the plate are digits. Each digit can range from 0 to 9, providing 10 choices per digit. Therefore, for three digits, the combinations are computed as \(10 \times 10 \times 10\).

Step 4: Multiply All Combinations

To find the total number of possible license plates, multiply the number of choices for each part of the plate: the non-zero digit, the three letters, and the three digits. This is calculated as \(9 \times 26^3 \times 10^3\).

Step 5: Compute Final Result

Calculate the expression \(9 \times 26^3 \times 10^3 = 9 \times 17,576 \times 1,000 = 158,184,000\). Therefore, there are 158,184,000 possible standard license plates in this system.

Key Concepts

Probability

The study of probability helps us understand how likely an event is to occur. In the context of license plates, it answers questions like, "How likely is it to randomly select a specific plate number?" To determine probability, we consider the number of favorable outcomes over the total number of possible outcomes.
For a specific plate, the probability is 1 out of the total number of combinations, which in this example is 158,184,000.

• Probability formula: \( P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}} \)
• Specialized for our license plate situation: \( P( ext{specific plate}) = \frac{1}{158,184,000} \)

As you can see, due to the large number of possibilities, the probability of any one specific plate is extremely low. This understanding of probability is fundamental to combinatorics, allowing us to quantify uncertainty and randomness in various scenarios.

Permutations and Combinations

Permutations and combinations are core concepts in combinatorics, dealing with the arrangement and selection of objects.

• Permutations: Refer to the arrangement of all members of a set into some sequence or order. Order matters here.
• Combinations: Focus on selecting members from a larger set. Order does not matter.

The problem of generating license plates is centered around permutations. Here, each section (non-zero digit, letters, digits) uses permutations to create distinct sequences. The order of numbers and letters on a plate contributes to its uniqueness.

• Example for letters: 26 letters permuted in three places: \(26 \times 26 \times 26\) permutations.
• Example for digits: In three positions, each with 10 possibilities: \(10 \times 10 \times 10\) permutations.

By multiplying the permutations, we find the total combinations possible in creating license plates.

Problem-Solving

Problem-solving often involves breaking down complex problems into smaller, manageable parts. The process of determining how many license plates can be made using specific rules is a perfect example. To solve the problem, you:

• Identify each component of the plate (non-zero digit, letters, digits) and list the possible choices for each.
• Apply the principle of multiplication to combine these choices, as independent events multiply the number of possibilities.
• Systematically compute the total number by calculating each part as done: \(9 \times 26^3 \times 10^3\).
• Simplify and calculate for the final answer of 158,184,000.

This methodology highlights the use of logic and arithmetic to solve real-world combinatorial problems. Structuring problems into clear steps is a cornerstone of successful problem-solving in mathematics.