License plates in the newly established state of Atlantis consist of three letters followed by three digits.  The first letter must be a vowel (y is not considered a vowel), and no letter can be repeated.  The first digit must be one of $\{1,2,4,6\}$, which corresponds to the location the license was issued.  No digit may be repeated.  How many possible license plates exist?
I need to count the number of ways to choose the letters and the digits, and then multiply them together, since they are independent choices. For the letters, I have five vowels to choose from for the first letter. Then, for the second letter, I have 25 remaining letters to choose from, since I can't repeat the first one. Similarly, for the third letter, I have 24 remaining letters to choose from, since I can't repeat the first or the second one. So, the number of ways to choose the letters is $5 \times 25 \times 24 = 3000$. For the digits, I have four choices for the first digit, based on the location. Then, for the second digit, I have nine remaining digits to choose from, since I can't repeat the first one. Similarly, for the third digit, I have eight remaining digits to choose from, since I can't repeat the first or the second one. So, the number of ways to choose the digits is $4 \times 9 \times 8 = 288$. Therefore, the total number of possible license plates is $3000 \times 288 = 864000$. 