  We present the building blocks that can be combined to produce tree-level
S-matrix elements of a variety of theories with various spins mixed in
arbitrary dimensions. The new formulas for the scattering of $n$ massless
particles are given by integrals over the positions of $n$ points on a sphere
restricted to satisfy the scattering equations. As applications, we obtain all
single-trace amplitudes in Einstein--Yang--Mills (EYM) theory, and
generalizations to include scalars. Also in EYM but extended by a B-field and a
dilaton, we present all double-trace gluon amplitudes. The building blocks are
made of Pfaffians and Parke--Taylor-like factors of subsets of particle labels.
