Final answer:
To calculate the number of different possible license plates, multiply the combinations for each position. For two letters followed by four digits, there are 6,760,000 possibilities. For three letters followed by three digits, there are 17,576,000 possibilities.
Step-by-step explanation:
The question concerns the concept of permutations and combinations in mathematics, specifically applied to creating license plates. We'll find out how many different license plate configurations can be created given certain rules about their format.
Part (a): Two Letters Followed by a Four-Digit Integer
In this case, since there are 26 possible letters for each of the two letter positions and 10 possible digits for each of the four numerical positions (0-9), we would calculate the total number of permutations by multiplying the possibilities for each position: 26 × 26 × 10 × 10 × 10 × 10. This equates to 6,760,000 different license plates possible.
Part (b): Three Letters Followed by a Three-Digit Integer
Similarly, with three letters and three numerical positions, the permutations are calculated as 26 × 26 × 26 × 10 × 10 × 10, yielding 17,576,000 different possible license plates, not accounting for any restricted spellings.
In practice, the actual number might be slightly less as some combinations of letters that form offensive words or unintended meanings are often avoided.
Please make sure to check these figures as needed to be sure they align with provided information encountered.