A [i]festive [/i] number is a four-digit integer containing one of each of the digits $0, 1, 2$, and $4$ in its decimal representation. How many festive numbers are there?
1. A festive number is a four-digit integer containing the digits $0, 1, 2,$ and $4$ exactly once. Since it is a four-digit number, the first digit cannot be $0$.
2. We need to determine the number of valid arrangements of the digits $0, 1, 2,$ and $4$ such that $0$ is not the leading digit.
3. First, we choose the position for the digit $0$. Since $0$ cannot be the first digit, it can only be in one of the last three positions (second, third, or fourth). Therefore, there are $3$ choices for the position of $0$.
4. After placing $0$, we have three remaining positions to fill with the digits $1, 2,$ and $4$. The number of ways to arrange these three digits in the remaining three positions is given by $3!$ (the factorial of $3$), which is $6$.
5. Therefore, the total number of festive numbers is the product of the number of ways to choose the position of $0$ and the number of ways to arrange the remaining digits:
   \[
   3 \times 3! = 3 \times 6 = 18
   \]

The final answer is $\boxed{18}$.