Problem 1509

1. Calculate the total number of possible license plates:
Each license plate consists of three letters followed by three digits.
- There are 26 possible choices for each letter.
- There are 10 possible choices for each digit.
Therefore, the total number of possible license plates is:
$$26^3 \times 10^3$$

2. Calculate the number of three-letter palindromes:
A three-letter palindrome has the form $ABA$, where $A$ and $B$ are letters.
- There are 26 choices for $A$.
- There are 26 choices for $B$.
Therefore, the number of three-letter palindromes is:
$$26 \times 26 = 676$$

3. Calculate the number of three-digit palindromes:
A three-digit palindrome has the form $CDC$, where $C$ and $D$ are digits.
- There are 10 choices for $C$.
- There are 10 choices for $D$.
Therefore, the number of three-digit palindromes is:
$$10 \times 10 = 100$$

4. Calculate the number of license plates with at least one palindrome:
- The number of license plates with a three-letter palindrome is:
$$676 \times 10^3$$
- The number of license plates with a three-digit palindrome is:
$$26^3 \times 100$$
- The number of license plates with both a three-letter and a three-digit palindrome is:
$$676 \times 100$$

Using the principle of inclusion and exclusion, the number of license plates with at least one palindrome is:
$$676 \times 10^3 + 26^3 \times 100 - 676 \times 100$$

5. Simplify the expression:
$$676 \times 10^3 + 26^3 \times 100 - 676 \times 100 = 676 \times 10^3 + 26^3 \times 100 - 676 \times 100$$
$$= 676 \times 10^3 + 26^3 \times 100 - 676 \times 100$$
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