2022 AMC 10A Problem 22

Let $n$ be the number of cards picked up on the first pass, where $1 \leq n \leq 12.$ If we choose the spaces that the $n$ cards occupy, the positions of the remaining cards are determined since they must be placed in order. There are $\binom{13}{n}$ ways to choose where the $n$ cards go, but if the $n$ cards are placed at the very beginning, then all the cards will be picked up on the first pass. Therefore, for a given $n$ there are $\binom{13}{n} - 1$ ways to arrange the cards. Summing over $1\le n\le12,$ we get $$\begin{gathered} \sum_{n=1}^{12}\left(\binom{13}{n}-1\right) \\ = (2^{13}-2)-12 \\ = 8178. \end{gathered}$$ Thus, D is the correct answer.