Set $B=\left\{ 1, 2, ... 18 \right\}$ is given. How many different operations $*$ can be defined so that $(B,*)$ is a groupoid with a property that $|\left\{i|i*(19-i) \neq i ∧ i*(19-i) \neq (19-i)\right\}| = k$, where $k \in B∪\left\{ 0 \right\}$.
What exactly is $k$ here? The solution is given as: $\binom{18}{k}(18-2)^k2^{18-k}18^{18^2-18}$.
I understand that the part within the parentheses say that eg. $18*1 \neq 1 ∧ \neq 18$ which explains the $18^{18^2-18}$ as a diagonal which follows these rules does not have 18 possibilities anymore. But what does this $k$ mean? And how does it change the result?
$k$ is the cardinality of the set of elements $i$ such that $i*(19-i)\notin\{i,19-i\}$; that is, it is the number of such elements.
Without constraints, there would be $18^{18^2}$ different operations, as we can independently assign one of $18$ results of the operation to each of the $18^2$ pairs of operands. Given the constraint on the cardinality of the set, we can choose the $k$ elements in the set in $\binom{18}k$ ways, assign one of $18-2$ admissible results to their products of the form $i*(19-i)$, yielding a factor $(18-2)^k$, and for the remaining $18-k$ elements we have to assign one of the two inadmissible results $i$ and $19-i$ to their products of the form $i*(19-i)$, yielding a factor $2^{18-k}$. That leaves $18^2-18$ products to be freely chosen among the $18$ possibilities, yielding a factor $18^{18^2-18}$.