  We analyse the full counting statistics (FCS) of the charge transport through
the Anderson impurity model (AIM) and similar systems with a single conducting
channel. The object of principal interest is the generating function for the
cumulants of charge current distribution. We derive an exact analytic formula
relating the FCS generating function to the self energy of the system in the
presence of the measuring field. We first check that our approach reproduces
correctly known results in simple limits, like the FCS of the resonant level
system (AIM without Coulomb interaction). We then proceed to study the FCS for
the AIM both perturbatively in the Coulomb interaction and in the Kondo regime
at the Toulouse point (we also study a related model of a spinless single-site
quantum dot coupled to two half-infinite metallic leads in the Luttinger liquid
phase at a special interaction strength). At zero temperature the FCS turns out
to be binomial for small voltages. For the generic case of arbitrary energy
scales the FCS is shown to be captured very well by generalisations of the
Levitov-Lesovik type formula. Surprisingly, the FCS for the AIM indicates a
presence of coherent electron pair tunnelling in addition to conventional
single-particle processes. By means of perturbative expansions around the
Toulouse point we succeeded in showing the universality of the binomial FCS at
zero temperature in linear response. Based on our general formula for the FCS
we then argue for a more general binomial theorem stating that the linear
response zero-temperature FCS for any interacting single-channel set-up is
always binomial.
