Distinct Partitions and Some q-Binomial Summation Identities

The partition functions $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, and $Q(n,m,p)$, the number of integer partitons of $n$ into exactly $m$ distinct parts with each part at most $p$, are related by double summation identities which follow from their generating functions. From these identities and some identities from an earlier paper, some other identities involving distinct partitions and some q-binomial summation identities are proved, and from these follow some combinatorial identities.