Homogeneity degree of some symmetric products

For a metric continuum X, we consider the nth-symmetric product Fn(X) defined as the hyperspace of all nonempty subsets of X with at most n points. The homogeneity degree hd(X) of a continuum X is the number of orbits for the action of the group of homeomorphisms of X onto itself. In this paper we determine hd(Fn(X)) for every manifold without boundary X and n∈N. We also compute hd(Fn[0,1]) for all n∈N.