might break up later on in the reaction process, and the two now disconnected members of the pair might find other partners. The initial number (molar amounts) of monomers of type A1, A2, and A3 are denoted by n [1], n[2], and n[3], respectively. Then, n tot = n[1] + n[2] + n [3] is the total number of monomeric units, andis the total number (molar amounts) of A groups available to form chemical bonds. The fraction of f tot is denoted by p (named degree of conversion, or extent of reaction) that has formed a bond. It is assumed that after a while, at each value of p, 0 ≤ p ≤ 1, the reaction process has reached a dynamic equilibrium in which the number of bonds does not change over time, and the total number of polymeric molecules remains at the stable value of n tot -1 2 p f tot . When a satisfactory level is reached, the reaction process is frozen by lowering the temperature and/or by removing the condensation by-product. In modeling the reaction process and the formation of polymeric molecules, we make the usual assumptions: i) all reactions occur with equal probability, independent of the size of the molecules involved and independent of the status of other A groups on the same monomeric unit; ii) cycles do not form, i.e., no reaction happens between A groups on the same molecule; and iii) there are no other reactions than between A groups. [1,2] The system "A1+A2+A3," that we study in this paper, is the most simple representative of the more general step-growth polymerizing systems of type "Afi," whereby the functionality (number of A groups) on each monomer species is not limited to three. It is our intention, in future studies, to extend the calculation method developed here for "A1+A2+A3" to step-growth polymerizing systems of type "AfiBgi." Those systems are characterized by monomers bearing A groups and/ or B groups; an A group reacts only with a B group, and vice versa.These step-growth polymerizing systems, "Afi" and "AfiBgi," and the many variations thereof are widely studied in the chemical literature and are of great interest to the polymer industry. Examples of commercially relevant systems "Afi" with AA condensation and water as condensation by-product, are: linear polymers, "A2": POM (polyoxymethylene of paraformaldehyde)Step-growth polymerizing systems of type "A1+A2+A3" are considered. The monomers bear one (A1), two (A2), or three (A3) identical reactive sites. In the reactor vessel, at a given degree of conversion, a wide range of polymeric molecules has formed, differing in both molecular size and in branching structure, determined by the laws of probability. In a slice of the molecular size distribution, all polymeric molecules have the same size (i.e., are built up by the same number of monomeric units), but differ in number and position of branching points (A3's). A method is presented to calculate the path length distribution for each such slice. Here, path length is the number of chemical bonds in the path connecting two monomeric units in the molecule. The shape and moments of this di...