  We present new, simple proofs for the enumeration of five of the ten symmetry
classes of plane partitions contained in a given box. Four of them are derived
from a simple determinant evaluation, using combinatorial arguments. The
previous proofs of these four cases were quite complicated. For one more
symmetry class we give an elementary proof in the case when two of the sides of
the box are equal. Our results include simple evaluations of the determinants
$\det(\delta_{ij}+{x+i+j\choose i})_{0\leq i,j\leq n-1}$ and
$\det({x+i+j\choose 2j-i})_{0\leq i,j\leq n-1}$, notorious in plane partition
enumeration, whose previous evaluations were quite intricate.
