65th Czech and Slovak Mathematical Olympiad

No two even numbers can be in the same set (pair). Let us call partitions of $\{1,2,\dots ,12\}$ with this property, that is one even and one odd number in each pair, even-odd partitions. The only further limitations are, that $6$ nor $12$ cannot be paired with $3$ or $9$, and $10$ cannot be paired with $5$.

That means, that odd numbers $1$, $7$ and $11$ can be paired with numbers $2$, $4$, $6$, $8$, $10$, $12$, numbers $3$ and $9$ with $2$, $4$, $8$, $10$ and number $5$ can be paired with $2$, $4$, $6$, $8$, $12$. We cannot use the product rule directly, we distinguish two cases: $5$ is paired with $6$ or $12$, in the second one $5$ is paired with one of $2$, $4$, and $8$. The possible pairings are $2\cdot 4\cdot 3\cdot 3\cdot 2\cdot 1=144$ in the first case, $3\cdot 3\cdot 2\cdot 3\cdot 2\cdot 1=108$ in the second case. Together $144+108=252$ pairings.