We have carried out a large grid of N-body simulations in order to investigate if mass loss as a result of primordial gas expulsion can be responsible for the large fraction of second-generation (SG) stars in globular clusters (GCs) with multiple stellar populations (MSPs). Our clusters start with two stellar populations in which 10 per cent of all stars are SG stars. We simulate clusters with different initial masses, different ratios of the half-mass radius of first to SG stars, different primordial gas fractions and Galactic tidal fields with varying strength. We then let our clusters undergo primordial gas loss and obtain their final properties such as mass, half-mass radius and the fraction of SG stars. Using our N-body grid we then perform a Monte Carlo analysis to constrain the initial masses, radii and required gas expulsion time-scales of GCs with MSPs. Our results can explain the present-day properties of GCs only if (1) a substantial amount of gas was present in the clusters after the formation of SG stars and (2) gas expulsion time-scales were extremely short (≲ 10⁵ yr). Such short gas expulsion time-scales are in agreement with recent predictions that dark remnants have ejected the primordial gas from GCs, and pose a potential problem for the asymptotic giant branch scenario. In addition, our results predict a strong anti-correlation between the number ratio of SG stars in GCs and the present-day mass of GCs. So far, the observational data show only a significantly weaker anti-correlation, if any at all.