  We reconsider the Boltzmann-Gibbs statistical ensemble in thermodynamics
using the multinomial coefficient approach. We show that an ensemble is defined
by the determination of four statistical quantities, the element probabilities
$p_i$, the configuration probabilities $P_j$, the entropy $S$ and the extremum
constraints (EC). This distinction is of central importance for the
understanding of the conditions under which a microcanonical, canonical and
macrocanonical ensemble is defined. These three ensembles are characterized by
the conservation of their sizes. A variation of the ensemble size creates
difficulties in the definitions of the quadruplet $\{p_i, P_j, S, \mt{EC}\}$,
giving rise for a generalization of the Boltzmann-Gibbs formalism, such one as
introduced by Tsallis. We demonstrate that generalized thermodynamics represent
a transformation of ordinary thermodynamics in such a way that the energy of
the system remains conserved.
  From our results it becomes evident that Tsallis's formalism is a very
specific generalization, however, not the only one. We also revisit the
Jaynes's Maximum Entropy Principle, showing that in general it can lead to
incorrect results and consider the appropriate corrections.
