Question

How many license plates can be made using two uppercase letters followed by a 3 -digit number?

Short answer

676,000 unique license plates can be made.

Step-by-step solution

Determine Possible Combinations for Letters

The license plate requires two uppercase letters. In the English alphabet, there are 26 letters. Therefore, for each of the two positions, there are 26 possible choices. The total number of combinations for the letters is calculated by multiplying the possible choices for each letter: 26 choices for the first letter and 26 choices for the second letter. Number of combinations for letters = 26 × 26 = 676.

Determine Possible Combinations for Digits

The license plate also requires a 3-digit number. Each digit can be any number from 0 to 9, giving us 10 possible choices for each digit. Therefore, the number of possible combinations for the 3-digit number is: 10 choices for the first digit, 10 choices for the second digit, and 10 choices for the third digit. Number of combinations for digits = 10 × 10 × 10 = 1000.

Calculate Total Number of License Plates

To find the total number of possible license plates, we need to multiply the number of combinations for the letters by the number of combinations for the digits. This gives us the total number of unique license plates that can be formed. Total number of license plates = 676 (letters) × 1000 (digits) = 676,000.

Key concepts

License Plates Permutations

When it comes to creating license plates, understanding permutations is essential. A permutation involves arranging items in a specific order. In our exercise, we need to form a license plate using two uppercase letters followed by three digits.

• Two Letter Positions: In the English alphabet, there are 26 letters. Since the order in which these letters appear matters, we can calculate the permutations as: \(26 \times 26 = 676\). This means we can create 676 different two-letter combinations.
• Three Digit Positions: For digits ranging from 0 to 9, and with three slots to fill, we also consider their arrangement important. Therefore, the possible permutations are: \(10 \times 10 \times 10 = 1000\).

By multiplying these two values, 676 for letters and 1000 for digits, we get a total of 676,000 possible unique license plates.

Combinations with Repetition

Combinations with repetition come into play when we select items from categories where the same item can be chosen more than once. In the context of creating license plates, repetition is vital.

• Letters: Even though there are 26 letters available, each letter can be repeated. For example, a license plate could be "AA123" or "ZZ999". This repetition allows us to multiply the 26 choices twice.
• Digits: Similarly, each digit can be selected repeatedly. A license plate can have any combination like "12", "34", or "56", repeated across the three slots, such as "111" or "999". This leads to the calculation of \(10 \times 10 \times 10 = 1000\) permutations for the digits.

Combining both the repeated choices from letters and numbers, we achieve the total number of license plates by multiplying the outcomes of each category, resulting in many unique plate options.

Discrete Mathematics Applications

Understanding license plate permutations is just one way discrete mathematics applies to real-world scenarios. Discrete mathematics is a core area of math dealing with distinct and separate values, including combinatorics.

• System Identification: License plates serve as a systematic way to uniquely identify vehicles. The permutations and combinations calculated here ensure each vehicle can have a distinct identifier.
• Efficiency in Design: Discrete mathematics helps in optimizing the license plate system to cover the maximum number of vehicles using minimal characters.
• Security and Validation: By leveraging combinatorial methods, authorities can ensure that the plates are not easily duplicated, enhancing vehicle security.

Through examples like these, the significance of discrete mathematics in engineering, computer science, transportation, and many other fields becomes clear, showing how mathematical theories are instrumental in creating efficient and effective systems.