Identities in combinatorics, I: on sorting two ordered sets

AbstractA binomial coefficient identity equivalent to Saalschutz's summation of a 3F2 hypergeometric series is proved combinatorially. The proof depends on the enumeration of ordered pairs (A,B) of subsets of{1,2,3,…,ν} in which |A|=n, |B|=m, and B contains exactly r elements of the first n elements of A∪B