We study the multiplicity distributions of events with hard jets in proton-proton collisions at LHC energies using PYTHIA 8 Monte-Carlo simulations. We demonstrate that the charged-hadron multiplicity distributions scale with jet momentum. This suggests that the Koba--Nielsen--Olesen (KNO) scaling holds within a jet. The in-jet scaling is fulfilled without multiple-parton interactions (MPI), but breaks down in case MPI is present without color reconnection. Our findings imply that KNO scaling is violated by parton shower or multiple-parton interactions in higher-energy collisions.