  The paper outlines an exact combinatorial approach to finite coagulating
systems. In this approach, cluster sizes and time are discrete, and the binary
aggregation alone governs the time evolution of the systems. By considering the
growth histories of all possible clusters, the exact expression is derived for
the probability of a coagulating system with an arbitrary kernel being found in
a given cluster configuration when monodisperse initial conditions are applied.
Then, this probability is used to calculate the time-dependent distribution for
the number of clusters of a given size, the average number of such clusters and
that average's standard deviation. The correctness of our general expressions
is proved based on the (analytical and numerical) results obtained for systems
with the constant kernel. In addition, the results obtained are compared with
the results arising from the solutions to the mean-field Smoluchowski
coagulation equation, indicating its weak points. The paper closes with a brief
discussion on the extensibility to other systems of the approach presented
herein, emphasizing the issue of arbitrary initial conditions.
