  Let $G$ be a finite group. We consider the problem of counting simultaneous
conjugacy classes of $n$-tuples and simultaneous conjugacy classes of commuting
$n$-tuples in $G$. Let $\alpha_{G,n}$ denote the number of simultaneous
conjugacy classes of $n$-tuples, and $\beta_{G,n}$ the number of simultaneous
conjugacy classes of commuting $n$-tuples in $G$. The generating functions
$A_G(t) = \sum_{n\geq 0} \alpha_{G,n}t^n,$ and $B_G(t) = \sum_{n\geq 0}
\beta_{G,n}t^n$ are rational functions of $t$. We show that $A_G(t)$ determines
and is completely determined by the class equation of $G$. We show that
$\alpha_{G,n}$ grows exponentially with growth factor equal to the cardinality
of $G$, whereas $\beta_{G,n}$ grows exponentially with growth factor equal to
the maximum cardinality of an abelian subgroup of $G$. The functions $A_G(t)$
and $B_G(t)$ may be regarded as combinatorial invariants of the finite group
$G$. We study dependencies amongst these invariants and the notion of
isoclinism for finite groups. We prove that the normalized functions
$A_G(t/|G|)$ and $B_G(t/|G|)$ are invariants of isoclinism families.
