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.
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