  We present a trajectory-based semiclassical calculation of the full counting
statistics of quantum transport through chaotic cavities, in the regime of many
open channels. Our method to obtain the $m$th moment of the density of
transmission eigenvalues requires two correlated sets of $m$ classical
trajectories, therefore generalizing previous works on conductance and shot
noise. The semiclassical results agree, for all values of $m$, with the
corresponding predictions from random matrix theory.
