  The exact solutions of a one-dimensional mixture of spinor bosons and spinor
fermions with $\delta$-function interactions are studied. Some new sets of
Bethe ansatz equations are obtained by using the graded nest quantum inverse
scattering method. Many interesting features appear in the system. For example,
the wave function has the $SU(2|2)$ supersymmetry. It is also found that the
ground state of the system is partial polarized, where the fermions form a spin
singlet state and the bosons are totally polarized. From the solution of Bethe
ansatz equations, it is shown that all the momentum, spin and isospin
rapidities at the ground state are real if the interactions between the
particles are repulsive; while the fermions form two-particle bounded states
and the bosons form one large bound state, which means the bosons condensed at
the zero momentum point, if the interactions are attractive. The charge, spin
and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz
equations are also derived and their solutions at some special cases are
obtained analytically.
