AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_19 / Suppose that 13 cards numbered 1,2,3,…,13 are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example …

Suppose that 13 cards numbered 1,2,3,…,13 are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1,2,3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7,8,9,10 on the fourth pass, and 11,12,13 on the fifth pass. For how many of the 13! possible orderings of the cards will the 13 cards be picked up in exactly two passes?

Suppose that $13$ cards numbered $1, 2, 3, \ldots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards $1, 2, 3$ are picked up on the first pass, $4$ and $5$ on the second pass, $6$ on the third pass, $7, 8, 9, 10$ on the fourth pass, and $11, 12, 13$ on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes?