Charged particle multiplicity distributions in positron-proton deep inelastic scattering at a centre-of-mass energy Formula omitted GeV are measured. The data are collected with the H1 detector at HERA corresponding to an integrated luminosity of 136 pb Formula omitted. Charged particle multiplicities are measured as a function of photon virtuality Formula omitted, inelasticity y and pseudorapidity Formula omitted 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 Formula omitted and the Bjorken variable Formula omitted are converted to the hadron entropy Formula omitted, and predictions from a quantum entanglement model are tested.