How Many License Plates Can Be Made with 3 Letters and 3 Numbers (Repetition Allowed)?

When designing a license plate system, one of the most critical considerations is the number of unique combinations that can be created. This is particularly important for legal and administrative purposes. In this article, we'll explore how to calculate the total number of license plates that can be made if both letters and numbers allow for repetition, using a specific example of 3 letters followed by 3 numbers.

Understanding the Problem

The task is to determine the number of unique license plates that can be created using 3 letters and 3 digits with repetition allowed. There are 26 letters in the English alphabet and 10 digits (0-9).

Step-by-Step Calculation

Step 1: Calculate Combinations for Letters

For the letters, we can select any of the 26 letters for each of the 3 positions. This is a simple multiplication problem:

26 choices for the first letter × 26 choices for the second letter × 26 choices for the third letter

This can be written as:

263 17,576

Step 2: Calculate Combinations for Numbers

For the numbers, we can select any of the 10 digits for each of the 3 positions:

10 choices for the first digit × 10 choices for the second digit × 10 choices for the third digit

This can be written as:

103 1,000

Step 3: Calculate the Total Number of License Plates

To find the total number of unique license plates, we multiply the number of letter combinations by the number of digit combinations:

17,576 (letter combinations) × 1,000 (digit combinations) 17,576,000

Alternative Permutation Calculations

As discussed earlier, if we consider the different permutations of the letters and numbers, the calculation becomes more complex. However, if we are dealing with a sequence of 6 characters where the first 3 are letters and the last 3 are digits, we can use permutations to calculate the total number of combinations.

First, calculate the number of ways to arrange the 3 letters and 3 digits:

The formula for permutations is:

(n!) / (n1! × n2! × ... × nk!)

Where n is the total number of items to arrange, and n1, n2, ... nk are the number of identical items among them.

In this case, we have 6 characters to arrange, with 3 letters and 3 digits:

(6!) / (3! × 3!) 120 / (6 × 6) 20

Now, multiply the number of permutations by the number of letter and digit combinations:

20 × 17,576 (letter combinations) × 1,000 (digit combinations) 351,520,000

Therefore, the total number of unique 6-character license plates (with 3 letters followed by 3 digits) is:

351,520,000

Conclusion

Whether using simple multiplication or complex permutation calculations, the total number of unique license plates that can be made with 3 letters and 3 numbers (with repetition allowed) is:

17,576,000

For more complex scenarios, the permutations approach can be applied to ensure all unique combinations are accounted for.