  We present the exact solution for the many-body wavefunction of a
one-dimensional mixture of bosons and spin-polarized fermions with equal masses
and infinitely strong repulsive interactions under external confinement. Such a
model displays a large degeneracy of the ground state. Using a generalized
Bose-Fermi mapping we find the solution for the whole set of ground-state
wavefunctions of the degenerate manifold and we characterize them according to
group-symmetry considerations. We find that the density profile and the
momentum distribution depends on the symmetry of the solution. By combining the
wavefunctions of the degenerate manifold with suitable symmetry and guided by
the strong-coupling form of the Bethe-Ansatz solution for the homogeneous
system we propose an analytic expression for the many-body wavefunction of the
inhomogeneous system which well describes the ground state at finite, large and
equal interactions strengths, as validated by numerical simulations.

Quantum Gases