text: The correct (extensive) formulas for systems A and B were obtained because we included all the possible ways that the two systems could exchange particles.  The use of combinations (i.e. N particles choose N) was used to ascertain the number of ways N particles can be divided into system A containing n particles and system B containing n particles.  This counting is not justified on physical grounds, but on the need to integrate over phase space.  As will be illustrated below, phase space contains not a single n-sphere and a single n-sphere, but instead
  pairs of n-spheres, all situated in the same -dimensional velocity space.  The integral over accessible phase space must include all of these n-spheres, as can be seen in the figure, which shows the actual velocity phase space associated with a gas that consists of three particles. Moreover, this gas has been divided into two systems, A and B.
  If we ignore the spatial variables, the phase space of a gas with three particles is three dimensional, which permits one to sketch the n-spheres over which the integral over phase space must be taken. If all three particles are together, the split between the two gases is 3|0.  Accessible phase space is delimited by an ordinary sphere (2-sphere) with a radius that is either  or  (depending which system has the particles).
  If the split is 2|1, then phase space consists of circles and points. Each circle occupies two dimensions, and for each circle, two points lie on the third axis, equidistant from the center of the circle.  In other words, if system A has 2 particles, accessible phase space consists of 3 pairs of n-spheres, each pair being a 1-sphere and a 0-sphere:
  Note that