We perform a population synthesis of protoplanetary discs including infall with a total of $50,000$ simulations using a 1D vertically integrated viscous evolution code, studying a large parameter space in final stellar mass. Initial conditions and infall locations are chosen based on the results from a radiation-hydrodynamic population synthesis of circumstellar discs. We also consider a different infall prescription based on a magnetohydrodynamic (MHD) collapse simulation in order to assess the influence of magnetic fields on disc formation. The duration of the infall phase is chosen to produce a stellar mass distribution in agreement with the observationally determined stellar initial mass function. We find that protoplanetary discs are very massive early in their lives. When averaged over the entire stellar population, the discs have masses of $sim 0.3$ and $0.1,mathrm{M_odot}$ for systems based on hydrodynamic or MHD initial conditions, respectively. In systems with final stellar mass $sim 1,mathrm{M_odot}$, we find disc masses of $sim 0.7,mathrm{M_odot}$ for the `hydro case and $sim 0.2,mathrm{M_odot}$ for the `MHD case at the end of the infall phase. Furthermore, the inferred total disc lifetimes are long, $approx 5-7,mathrm{Myr}$ on average, despite our choice of a high value of $10^{-2}$ for the background viscosity $alpha$-parameter. In addition, fragmentation is common in systems that are simulated using hydrodynamic cloud collapse, with more fragments of larger mass formed in more massive systems. In contrast, if disc formation is limited by magnetic fields, fragmentation is suppressed entirely.
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