  In this note we give an algorithm to determine the rational homotopy type of
the free and pointed mapping spaces $ map(F(\mathbb R^m,k), S^n)$ and $
map^*(F(\mathbb R^m,k), S^n)$. An explicit description of these spaces is given
for $k=3$. The general case for $n$ odd is also presented as an immediate
consequence of the rational version of a classical result of Thom.
