Among the many reasons underlying the practice of randomization some of the main ones can be described asaveragingout oreliminationof the effects of nuisance parameters. It is already well known (Godambe 1966) that averaging over all the possible results of the adopted randomization is directly in conflict with thelikelihood principle.The process of elimination of nuisance parameters has possibly deeper intuitive appeal. But this process contradicts a very basic principle of statistical inference subsequently defined as theancillarity principle.This is demonstrated in relation to a practice of randomization calledbalanced sampling.We would use a formalism very similar to that of Birnbaum (1962).