HMMT November 2021 U24

Find the number of subsets $S$ of $\{1,2, \ldots, 48\}$ satisfying both of the following properties:

• For each integer $1 \leq k \leq 24$, exactly one of $2 k-1$ and $2 k$ is in $S$.
• There are exactly nine integers $1 \leq m \leq 47$ so that both $m$ and $m+1$ are in $S$.