Digit Set - The Setup

You can only use digits from the given set. At each position, instead of trying $0$-$9$, you try only the allowed digits. observation: numbers shorter than $n$ are always valid (if they use allowed digits).

A 2-digit number made from $[1,3,5,7]$ is $\leq 100$. For numbers with the same length as $n$, you need the tight bound logic. Can you place digit $d$ at position $pos$ without exceeding $n$? Transform the problem: count valid numbers by building them digit by digit from left to right.