There are $100$ domino tiles $\{i,j\}$ with $i,j \in \{1, 2,\dots, 25\}$ and $i\neq j$ in a sack, with no domino tile occurring twice. Each number $1, 2,\dots, 25$ is on exactly $8$ dominoes.
How may ways to pick $n \leq 100$ dominoes, such that each number $1, 2,\dots, 25$ is an even number of times in the selection?
So for example for $n = 3$, a valid selection would be $\{\{1,2\},\{1,3\},\{2,3\}\}$ if these dominoes were in the sack.
Or is there an efficient algorithm to answer the above question, in the case that the contents of the sack are known?