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 A) was used to ascertain the number of ways N particles can be divided into system A containing n A particles and system B containing n B 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 A -sphere and a single n B -sphere, but instead