Combinatorics Is Fun

Stack has $N$ distinct cards, numbered $1$ to $N$. He wonders how many pairs of arrays $(A,B)$ are there such that: - $|A|=|B|=N$; - $1 \leq A_i, B_i \leq N$; - $A_i \neq B_i$; - If he writes $A_i$ on the upper face of $i^{th}$ card and $B_i$ on the lower face of $i^{th}$ card, it is possible to flip some of the cards in such a way that each integer from $1$ to $N$ is present on the upper face of