  Let $F$ be a graph which contains an edge whose deletion reduces its
chromatic number. We prove tight bounds on the number of copies of $F$ in a
graph with a prescribed number of vertices and edges. Our results extend those
of Simonovits, who proved that there is one copy of $F$, and of Rademacher,
Erd\H os and Lov\'asz-Simonovits, who proved similar counting results when $F$
is a complete graph.
  One of the simplest cases of our theorem is the following new result. There
is an absolute positive constant $c$ such that if $n$ is sufficiently large and
$1 \le q < cn$, then every $n$ vertex graph with $n$ even and $n^2/4 +q$ edges
contains at least $q(n/2)(n/2-1)(n/2-2)$ copies of a five cycle. Similar
statements hold for any odd cycle and the bounds are best possible.
