We know that algebraic operation is a function $f:\underbrace{\left ( X\times X\times \cdots \times X\right )}_{t\ \text{times}}\rightarrow{X}$
If $X$ is a set and and cardinality is $|X|=n$ then there are $n^{(n^t)}$ functions or 't'-nary algebraic operations on that set.
If $t=2$ then the operation is called a "binary operation" and there are $n^{(n^2)}$ operations.
If cardinality of $|X|=2$ then we will have $2^{(2^2)}=16$ binary operations.
Those operations are:
$$1)
\begin{array}{c|ccccc}
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& a& a& \\
& a& a&
\end{array}
2)
\begin{array}{c|ccccc}
* & & \\ \hline
& a& a& \\
& a& b&
\end{array}
3)
\begin{array}{c|ccccc}
* & & \\ \hline
& a& a& \\
& b& a&
\end{array}
4)
\begin{array}{c|ccccc}
* & & \\ \hline
& a& b& \\
& a& a&
\end{array}
5)
\begin{array}{c|ccccc}
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& a& a&
\end{array} $$
$$6)
\begin{array}{c|ccccc}
* & & \\ \hline
& a& a& \\
& b& b&
\end{array}
7)
\begin{array}{c|ccccc}
* & & \\ \hline
& a& b& \\
& a& b&
\end{array}
8)
\begin{array}{c|ccccc}
* & & \\ \hline
& b& a& \\
& a& b&
\end{array}
9)
\begin{array}{c|ccccc}
* & & \\ \hline
& a& b& \\
& b& a&
\end{array}
10)
\begin{array}{c|ccccc}
* & & \\ \hline
& b& a& \\
& b& a&
\end{array}
11)
\begin{array}{c|ccccc}
* & & \\ \hline
& b& b& \\
& a& a&
\end{array} $$
$$12)
\begin{array}{c|ccccc}
* & & \\ \hline
& a& b& \\
& b& b&
\end{array}
13)
\begin{array}{c|ccccc}
* & & \\ \hline
& b& a& \\
& b& b&
\end{array}
14)
\begin{array}{c|ccccc}
* & & \\ \hline
& b& b& \\
& a& b&
\end{array}
15)
\begin{array}{c|ccccc}
* & & \\ \hline
& b& b& \\
& b& a&
\end{array}
16)
\begin{array}{c|ccccc}
* & & \\ \hline
& b& b& \\
& b& b&
\end{array}
$$
$$a,b \in \mathbb X$$
Only 8 operations from this list are associative. Right!
Those are $1, 2, 6, 7, 8, 9, 12, 16$
Because they satisfy associativity condition: $(a*b)*c=a*(b*c)$ where $a,b,c \in \mathbb X$.
Things will get more difficult if $n=3$ or $|X|=3$, we will have 113 associative operations.
But what if $t=3$?
$$f:X\times X\times X \rightarrow{X}$$
Then the operation is called a "ternary operation" and there are $n^{(n^3)}$ operations. Right!
My question is How many ternary operations from that are associative?
Even when cardinality is $|X|=2$, How many associative operations might be from $2^{(2^3)}=2^8=256$ operations?
And finally: Is there a formula about computing ternary operations on a finite set with $|X|=n$ cardinality?