To calculate the number of different license plates that could be created with at least one letter or digit repeated, we can use the principle of inclusion-exclusion.
Step 1: Calculate the total number of possible license plates:
Since the license plate consists of 2 letters followed by 4 digits, there are 26 letters in the English alphabet (excluding the letters I and O) and 10 digits (0-9). This gives us a total of (26 x 26) x (10 x 10 x 10 x 10) = 676,000.
Step 2: Calculate the number of license plates with no repeated letters or digits:
To calculate this, we need to determine the number of ways to choose 2 distinct letters from 26, which is given by the combination formula C(26,2) = 26! / (2!(26-2)!) = 325. Similarly, there are 10 choices for each digit, so the total number of license plates without repeated letters or digits is 325 x 10 x 10 x 10 x 10 = 3,250,000.
Step 3: Calculate the number of license plates with no repeated letters:
To calculate this, we need to determine the number of ways to choose 2 distinct letters from 26, as we did in Step 2. Thus, the number of license plates without repeated letters is 325 x 10 x 10 x 10 x 10 = 3,250,000.
Step 4: Calculate the number of license plates with no repeated digits:
Similar to the previous step, we now need to determine the number of ways to choose 2 distinct digits from 10, which is given by the combination formula C(10,2) = 10! / (2!(10-2)!) = 45. Thus, the number of license plates without repeated digits is 45 x 26 x 26 = 30,120.
Step 5: Apply the inclusion-exclusion principle:
Now, we can calculate the number of license plates that have at least one letter or digit repeated by subtracting the number of license plates without repeated letters (Step 3) and the number of license plates without repeated digits (Step 4) from the total number of possible license plates (Step 1). Thus, the number of license plates with at least one letter or digit repeated is 676,000 - 3,250,000 - 30,120 = 642,880.
Therefore, there are 642,880 different license plates that could be created with at least one letter or digit repeated.