  For various triple systems $F$, we give tight lower bounds on the number of
copies of $F$ in a triple system with a prescribed number of vertices and
edges. These are the first such results for hypergraphs, and extend earlier
theorems of Bollob\'as, Frankl, F\"uredi, Keevash, Pikhurko, Simonovits, and
Sudakov who proved that there is one copy of $F$.
  A sample result is the following: F\"uredi-Simonovits and independently
Keevash-Sudakov settled an old conjecture of S\'os by proving that the maximum
number of triples in an $n$ vertex triple system (for $n$ sufficiently large
and even) that contains no copy of the Fano plane is $p(n)={n/2 \choose 2}n.$
  We prove that there is an absolute constant $c$ such that if $n$ is
sufficiently large and $1 \le q \le cn^2$, then every $n$ vertex triple system
with $p(n)+q$ edges contains at least $6q({n/2 \choose 4}+(n/2 -3){n/2 \choose
3}$$ copies of the Fano plane. This is sharp for $q\le n/2-2$.
  Our proofs use the recently proved hypergraph removal lemma and stability
results for the corresponding Tur\'an problem.
