Problem 28 - Algebra and Trigonometry

Question

A state license plate design has six places. Each plate begins with a fixed number of letters, and the remaining places are filled with digits. (For example, one letter followed by five digits, two letters followed by four digits, and so on.) The state has 17 million registered vehicles. (a) The state decides to change to a system consisting of one letter followed by five digits. Will this design allow for enough different plates to accommodate all the vehicles registered? (b) Find a system that will be sufficient if the smallest possible number of letters is to be used.

Step-by-Step Solution

Answer

(a) No, (b) Three letters and three digits allow for sufficient plates.

Step 1: Determine the number of plate combinations for one letter and five digits

Each letter has 26 possible choices (A to Z), and each digit has 10 possible choices (0 to 9). Therefore, the number of possible license plates with one letter followed by five digits is calculated as \(26 \times 10^5\). This equals \(26 \times 100,000 = 2,600,000\) possible plates.

Step 2: Compare number of plates to registered vehicles for part (a)

The state has 17 million registered vehicles. With only 2,600,000 possible plates (one letter followed by five digits) available, 17 million vehicles far exceed the capacity of this plate design.

Step 3: Calculate combinations with an additional letter

To accommodate more vehicles, let's try two letters followed by four digits. The number of possible plates is \(26^2 \times 10^4\). That's \(26 \times 26 \times 10,000 = 6,760,000\) plates, which is still fewer than 17 million.

Step 4: Try three letters with three digits

Calculate the combinations for three letters followed by three digits. The total is \(26^3 \times 10^3\). This equals \(26 \times 26 \times 26 \times 1,000 = 17,576,000\), which covers all 17 million registered vehicles.

Step 5: Validate the solution system

With three letters and three digits, there are enough plates to accommodate all current vehicles. Ensure the state system allows such combinations for implementation.

Key Concepts

license plate combinations

Understanding how license plate combinations are formed is crucial in determining whether a specific plate design can serve all registered vehicles. A standard approach is to decide on the number of letters and digits in the license plate sequence. For instance, a combination of one letter followed by five digits offers a distinct set of possibilities. Each character in the plate can be varied in specific ways:

• Letters: 26 options (A to Z)
• Digits: 10 options (0 to 9)

The formula to calculate the total combinations is given by multiplying the choices for letters and digits. As demonstrated in our exercise, one letter followed by five digits is calculated as:\[26 \times 10^5 = 2,600,000 \]This indicates the maximum distinctive plates possible with this configuration.

combinatorial counting

Combinatorial counting is a fundamental concept in understanding how to compute the number of different combinations or arrangements possible. In the context of license plates, combinatorial counting helps in determining how many unique plates can be created based on the arrangement of letters and digits.
When dealing with combinations like in our license plate problem, it's crucial to:

• Identify the number of positions or places (e.g., six in this case).
• Determine the possible options for each position. For letters, there are 26 possibilities, and for digits, there are 10.

By multiplying these options together, we can find the number of potential plates. This technique enables state authorities to forecast whether a proposed design will accommodate all vehicles.

permutations and combinations

The study of permutations and combinations is essential in solving problems like license plate generation. While these two concepts are sometimes used interchangeably, they refer to different types of counting problems:

• Permutations consider the arrangement of items where the order matters. When arranging letters and numbers in a plate, permutations play a role in deciding sequence possibilities.
• Combinations focus on grouping without regard to the order. For license plate combinations, we're primarily interested in permutations since the sequence of letters and digits impacts the final count.

For example, with a setup of three letters and three digits, the formula is:\[26^3 \times 10^3 = 17,576,000\]This assures us that all possible permutations cover the 17 million registered vehicles, allowing an adequate design solution.

problem-solving strategies

Developing effective problem-solving strategies can greatly facilitate finding solutions to combinatorial challenges such as license plate designs. Our approach to determining a sufficient system involves:

• Identifying Constraints: Recognizing the number of registered vehicles (17 million in this scenario).
• Evaluating Designs: Testing various configurations (single letter vs. multiple letters) to meet vehicle requirements.
• Increasing Complexity as Needed: Moving from one letter to three ensures adequate combinations.
• Verification: Recalculating to confirm the solution satisfies all constraints.

By employing these strategies, we ensure that the state implements a license plate system capable of serving its needs effectively.