  The time evolution of a system of coagulating particles under the product
kernel and arbitrary initial conditions is studied. Using the improved
Marcus-Lushnikov approach, the master equation is solved for the probability
$W(Q,t)$ to find the system in a given mass spectrum
$Q=\{n_1,n_2,\dots,n_g\dots\}$, with $n_g$ being the number of particles of
size $g$. The exact expression for the average number of particles, $\langle
n_g(t)\rangle$, at arbitrary time $t$ is derived and its validity is confirmed
in numerical simulations of several selected initial mass spectra.
