On the basis of equations obtained in the framework of second-order quantum-mechanical perturbation theory, the standard approach to the calculation of scattering radiation probability is extended to the case of ultrashort laser pulses. We investigate the mechanism of the appearance of plasmon peaks in the spectrum of the plasma form factor for different parameters of the problem. For the case in which scattering on plasmons dominates over scattering on electron density fluctuations caused by chaotic thermal motion, we derive analytical expressions describing the scattering probability of ultrashort laser pulses on plasmons. Together with this, we obtain a simple expression connecting the frequency of scattered radiation and the energy transmitted from the incident pulse to plasmon, and vice versa. In considering the scattering probability, our emphasis is on the dependence on the pulse duration. We assess in detail the trends of this dependence for various relations between pulse carrier frequency and plasmon energy.