A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $2$-player teams, but no teams may have exactly the same $2$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are among those $9$ people, is equal to $1$. How many values $n$, $9\le n\le 2017$, can be the number of participants?