  We consider testing independence in group-wise selections with some
restrictions on combinations of choices. We present models for frequency data
of selections for which it is easy to perform conditional tests by Markov chain
Monte Carlo (MCMC) methods. When the restrictions on the combinations can be
described in terms of a Segre-Veronese configuration, an explicit form of a
Gr\"obner basis consisting of moves of degree two is readily available for
performing a Markov chain. We illustrate our setting with the National Center
Test for university entrance examinations in Japan. We also apply our method to
testing independence hypotheses involving genotypes at more than one locus or
haplotypes of alleles on the same chromosome.
