  Charged particle multiplicity distributions in positron-proton deep inelastic
scattering at a centre-of-mass energy $\sqrt{s}=319$ GeV are measured. The data
are collected with the H1 detector at HERA corresponding to an integrated
luminosity of $136$ pb${}^{-1}$. Charged particle multiplicities are measured
as a function of photon virtuality $Q^2$, inelasticity $y$ and pseudorapidity
$\eta$ in the laboratory and the hadronic centre-of-mass frames. Predictions
from different Monte Carlo models are compared to the data. The first and
second moments of the multiplicity distributions are determined and the KNO
scaling behaviour is investigated. The multiplicity distributions as a function
of $Q^2$ and the Bjorken variable $x_{\rm Bj}$ are converted to the hadron
entropy $S_{\rm hadron}$, and predictions from a quantum entanglement model are
tested.
