Using the wave functions of the multichannel-quantum-defect theory, a semiclassical theory for both elastic and inelastic collisional cross sections of Rydberg atoms, in which there can be more than one valence electron, with ground-state rare-gas atoms is obtained. Especially the noble-gas-induced broadening cross sections of Rydberg series in alkaline-earth atoms can be calculated. An application of theory is shown to explain the irregular broadening rates of Sr 5{ital sns} ¹{ital S}â series due to collision with Xe, and a good agreement between theory and experiment is achieved.