We explore the effects of progenitor pre-stellar core properties on the evolution of disks with external photoevaporation in clusters. Since the strength of external photoevaporation is largely determined by the depth of the gravitational potential well of the disk, the external photoevaporation rate is the function of star mass and disk size. The properties of a core collapse set up the initial conditions of protoplanetary disks, so they influence the evolutions of star mass and disk size. Our calculations show that the core properties can dramatically influence the efficiency of external photoevaporation. For the core with low angular velocity, most core mass directly falls onto the central star or onto the disk near the star. External photoevaporation is suppressed even if external radiation from nearby massive stars are strong. In this case, the disk evolution in clusters is primarily driven by its own internal viscosity. However, if the core angular velocity is high, most core mass falls onto the disk far from the central star. External photoevaporation is so strong that the disk mass is severely evaporated. Finally, the star mass is very low and the disk lifetime is very short. Our calculations could interpret some observational features of disks in clusters, such as the diameter distribution of disks in the Trapezium cluster and the correlation between mass accretion rate and star mass. We suggest that the disk mass determined by (sub) millimeter wavelength observations may be underestimated.