A K-matrix solution to the coupled, inhomogeneous equations describing the scattering of a particle by a system of identical particles is developed. It is shown that K is a sum of two terms, one arising from the homogeneous solution and one from the particular integral. The former is a direct contribution, i.e., with no exchange, while the latter is a pure exchange contribution. Thus, as in the previously studied case of the T matrix arising from this system of equations, the direct and exchange portions of K are additive, and can be computed separately. A unitary S matrix is obtained from K in the usual way: S = (1 + iK)(1 − iK) −1. The problem of how to calculate K when an apparent two-channel problem is actually a two-particle problem with the channels referring to the identical particle labels is also solved.