A lock requires a 4-character password formed using digits $0-9$ and letters $A-F$, with no repetition. How many such passwords are possible?

There are $10$ digits from $0$ to $9$ and $6$ letters from $A$ to $F$, making a total pool of $10+6=16$ characters.
Since repetition is not allowed, the number of ways to arrange $4$ characters out of $16$ is given by the permutation:

$${}^{16}{P}_4=16\times 15\times 14\times 13=43680$$