  We give "hybrid" proofs of the $q$-binomial theorem and other identities. The
proofs are "hybrid" in the sense that we use partition arguments to prove a
restricted version of the theorem, and then use analytic methods (in the form
of the Identity Theorem) to prove the full version. We prove three somewhat
unusual summation formulae, and use these to give hybrid proofs of a number of
identities due to Ramanujan. Finally, we use these new summation formulae to
give new partition interpretations of the Rogers-Ramanujan identities and the
Rogers-Selberg identities.
