  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.
