Charged particle multiplicity distributions in positron-proton deep inelastic scattering at a centre-of-mass energy 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 Q2, inelasticity y and pseudorapidity η 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 Q2 and the Bjorken variable xbj are converted to the hadron entropy Shadron, and predictions from a quantum entanglement model are tested.