The multiple scattering theory of Kerman, McManus, and Thaler is extended to include triple scattering effects. With a judicious choice of the propagator between scatterings the triple scattering term is approximately proportional to the triple correlation function. The methods of Feshbach, Gal, and Hufner are generalized to provide a coupled channel description of the elastic scattering of fast particles including triple scattering. The coupling potentials depend directly on the pair and triple correlation functions. Numerical calculations are performed for protons of 1.69 BeV/c momentum on 4He. Center-of-mass correlations and phenomenological short-range dynamical correlations are tested using a spin isospin averaged nucleon-nucleon amplitude. The differential cross section around the second diffraction minimum and third diffraction peak is particularly sensitive to triple scattering effects. Present experimental accuracy around the third diffraction peak does not permit any definitive conclusions to be made.