  We develop a general procedure that finds recursions for statistics counting
isomorphic copies of a graph $G_0$ in the common random graph models ${\cal
G}(n,m)$ and ${\cal G}(n,p)$. Our results apply when the average degrees of the
random graphs are below the threshold at which each edge is included in a copy
of $G_0$. This extends an argument given earlier by the second author for
$G_0=K_3$ with a more restricted range of average degree. For all strictly
balanced subgraphs $G_0$, our results gives much information on the
distribution of the number of copies of $G_0$ that are not in large "clusters"
of copies. The probability that a random graph in ${\cal G}(n,p)$ has no copies
of $G_0$ is shown to be given asymptotically by the exponential of a power
series in $n$ and $p$, over a fairly wide range of $p$. A corresponding result
is also given for ${\cal G}(n,m)$, which gives an asymptotic formula for the
number of graphs with $n$ vertices, $m$ edges and no copies of $G_0$, for the
applicable range of $m$. An example is given, computing the asymptotic
probability that a random graph has no triangles for $p=o(n^{-7/11})$ in ${\cal
G}(n,p)$ and for $m=o(n^{15/11})$ in ${\cal G}(n,m)$, extending results of the
second author.
