Question 82
Compute the number of triples $(f, g, h)$ of permutations on $\{1,2,3,4,5\}$ such that
$$
\begin{aligned}
& f(g(h(x)))=h(g(f(x)))=g(x), \\
& g(h(f(x)))=f(h(g(x)))=h(x), \text{ and } \\
& h(f(g(x)))=g(f(h(x)))=f(x)
\end{aligned}
$$
for all $x \in \{1,2,3,4,5\}$.
$$
\begin{aligned}
& f(g(h(x)))=h(g(f(x)))=g(x), \\
& g(h(f(x)))=f(h(g(x)))=h(x), \text{ and } \\
& h(f(g(x)))=g(f(h(x)))=f(x)
\end{aligned}
$$
for all $x \in \{1,2,3,4,5\}$.