  For various quadruple systems F, we give asymptotically sharp lower bounds on
the number of copies of F in a quadruple system with a prescribed number of
vertices and edges. Our results extend those of Furedi, Keevash, Pikhurko,
Simonovits and Sudakov who proved under the same conditions that there is one
copy of $F$. Our proofs use the hypergraph removal Lemma and stability results
for the corresponding Turan problem proved by the above authors.
