  We principally present reductions of certain generalized hypergeometric
functions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most of
these results have been known for some time, but one of the methods, wherein we
simultaneously solve for three alternating binomial sums, may be new. We obtain
a functional equation holding for all three of this set of alternating binomial
sums. Using successive derivatives, we show how related chains of $_3F_2(\pm
1)$ values may be obtained. It may be emphasized that we make no reliance on
the WZ method for hypergeometric summation. Additional material on Pochhammer
symbols and certain of their products is presented in an Appendix to supplement
the pedagogical content of the paper.
