Problem 29 - Algebra and Trigonometry

Question

License Plates A state has registered 8 million automobiles. To simplify the license plate system, a state employee suggests that each plate display only two letters followed by three digits. Will this system create enough different license plates for all the vehicles that are registered?

Step-by-Step Solution

Answer

No, the system can create only 676,000 plates, not enough for 8 million vehicles.

Step 1: Understand the Problem

We need to determine if a system of license plates with 2 letters followed by 3 digits can produce at least 8 million unique combinations, as there are 8 million registered automobiles.

Step 2: Calculate Combinations of Letters

Each position in the letter part of the plate can be one of 26 letters in the alphabet. Thus, for 2 letters, the total number of combinations is calculated as \(26 \times 26 = 676\).

Step 3: Calculate Combinations of Digits

Each digit can be one of 10 numbers (0 through 9). Therefore, for 3 digits, the total number of combinations is \(10 \times 10 \times 10 = 1000\).

Step 4: Calculate Total Combinations

The total number of different license plates that can be created is the product of the combinations of the letters and digits: \(676 \times 1000 = 676,000\).

Step 5: Compare with Required Number

We compare the number of possible license plate combinations (676,000) with the number of registered automobiles (8,000,000). Since 676,000 is much less than 8,000,000, there are not enough unique license plates available under this system.

Key Concepts

License Plates

License plates are identifiers assigned to vehicles, and they serve to uniquely distinguish each registered car within a state or country. These plates often blend both letters and numbers to create unique combinations. In this exercise, we're tasked with evaluating a proposed license plate system of having two letters followed by three digits. This design intends to generate enough unique combinations to cover all registered vehicles. The goal is to verify whether this setup can provide at least 8 million different plates.

Unique Combinations

The heart of this problem lies in determining the number of unique combinations that license plates can display. A unique combination means each vehicle gets a distinct plate, with no repeats across the registry. The key steps involve:

• Calculating combinations of letters - each letter can be one of 26 options, hence two letters allow for 26 multiplied by 26 combinations.
• Calculating combinations of numbers - each digit can be selected from 0 to 9, resulting in 10 options per digit and three digits create 1000 combinations.
• Multiplying these results - the total combinations equal 676 letter combinations times 1000 digit combinations, leading to 676,000 unique license plates.

Comparing this against 8 million cars shows the proposed system falls short, highlighting the complexity and importance of adequate unique identifiers.

Alphabet Permutations

Permutations in the context of the alphabet refer to the different ways to arrange a given set of letters. With 26 English alphabet letters available, if we use two letters on the license plate, it helps to calculate permutations. This is seen through:

• For one spot, there are 26 possible choices, and for two spots, you multiply 26 by 26, providing 676 permutations.
• This limited space shows quickly how the permutation concept influences the total unique outcomes available.

This understanding helps to appreciate why careful consideration of character spaces in system design is crucial.

Number System

The number system plays a crucial role in determining the feasibility of the license plate design. Numbers have vast implied utility given each digit's independent nature and the limited range (0-9).

• In our setting, three digits expand to 10 options per digit, leading to a setup where the sequence permits 1000 combinations (10 x 10 x 10).
• This multiplication showcases the potential of the decimal system in helping create substantial variations, albeit not enough in this case.

A profound understanding of how the number system pairs with alphabets enriches the strategy, potentially prompting mixed-system designs with optimization goals in mind.