To solve this problem, let's break it down step by step:

Step 1: Total Number of License Plates Without Restrictions

A license plate consists of two letters followed by three digits. Let's calculate the total number of possible license plates without any restrictions:

• For the letters: There are 26 possible choices for each letter (since there are 26 letters in the English alphabet). So, for the two letters, the total number of combinations is: $26 \times 26 = 676$

• For the digits: There are 10 possible choices for each digit (from 0 to 9). So, for the three digits, the total number of combinations is: $10 \times 10 \times 10 = 1000$

Therefore, the total number of license plates without any restrictions is: $676 \times 1000 = 676,000$

Step 2: Number of License Plates with No Repeats (No Letters or Digits Repeated)

Now, we will calculate the number of license plates where no letter or digit is repeated:

• For the letters: There are 26 choices for the first letter, but only 25 choices for the second letter (since it cannot be the same as the first one). Therefore, the number of ways to choose the two letters with no repeats is: $26 \times 25 = 650$

• For the digits: There are 10 choices for the first digit, 9 choices for the second digit (it can't repeat the first digit), and 8 choices for the third digit (it can't repeat the first or second digit). So, the number of ways to choose the three digits with no repeats is: $10 \times 9 \times 8 = 720$

Thus, the number of license plates with no repeats is: $650 \times 720 = 468,000$

Step 3: License Plates with at Least One Repeated Letter or Digit

To find the number of license plates with at least one repeated letter or digit, we subtract the number of plates with no repeats from the total number of plates:

$676,000 - 468,000 = 208,000$

Final Answer:

So, the number of license plates that can be created with at least one repeated letter or digit is: $\boxed{208,000}$

Would you like any further clarification or additional details on this problem?

Here are some related questions for deeper understanding:

1. What is the difference between permutations and combinations, and how does it apply to this problem?
2. How would the calculation change if the number of letters increased to 30 instead of 26?
3. What if the license plate had four digits instead of three? How would the total number change?
4. How do you calculate the probability of getting a license plate with no repeats?
5. How can you apply the inclusion-exclusion principle in cases with more than two types of characters (letters and digits)?

Tip: When calculating probabilities or counts of repeated and non-repeated outcomes, always start by calculating the total possible outcomes, then subtract the cases that meet the conditions you want to exclude.