  A $d$-ary quasigroup of order $n$ is a $d$-ary operation over a set of
cardinality $n$ such that the Cayley table of the operation is a
$d$-dimensional latin hypercube of the same order. Given a binary quasigroup
$G$, the $d$-iterated quasigroup $G^{\left[d\right]}$ is a $d$-ary quasigroup
that is a $d$-time composition of $G$ with itself. A $k$-multiplex (a $k$-plex)
$K$ in a $d$-dimensional latin hypercube $Q$ of order $n$ or in the
corresponding $d$-ary quasigroup is a multiset (a set) of $kn$ entries such
that each hyperplane and each symbol of $Q$ is covered by exactly $k$ elements
of $K$. A transversal is a 1-plex.
  In this paper we prove that there exists a constant $c(G,k)$ such that if a
$d$-iterated quasigroup $G$ of order $n$ has a $k$-multiplex then for large $d$
the number of its $k$-multiplexes is asymptotically equal to $c(G,k)
\left(\frac{(kn)!}{k!^n}\right)^{d-1}$. As a corollary we obtain that if the
number of transversals in the Cayley table of a $d$-iterated quasigroup $G$ of
order $n$ is nonzero then asymptotically it is $c(G,1) n!^{d-1}$.
  In addition, we provide limit constants and recurrence formulas for the
numbers of transversals in two iterated quasigroups of order 5, characterize a
typical $k$-multiplex and estimate numbers of partial $k$-multiplexes and
transversals in $d$-iterated quasigroups.
