  We study a mixture of $N_{b}$ bosons with point hard-core boson-boson
interactions and $N_{f}$ noninteracting spinless fermions with point hard-core
boson-fermion interactions in 1D optical lattice with external harmonic confine
potential. Using an extended Jordan-Winger transformation (JWT) which maps the
hard-core Bose-Fermi mixture into two component noninteracting spinless
fermions with hard-core interactions between them, we get the ground states of
the system. Then we determine in details the one particle density matrix,
density profile, momentum distribution, the natural orbitals and their
occupations based on the constructed ground state wavefunctions. We also
discuss the ground state properties of the system with large but finite
interactions which lead to the lift of ground degeneracy. Our results show
that, although the total density profile is almost not affected, the
distributions for bosons and fermions strongly depend on the relative
strengthes of boson-boson interactions and boson-fermion interactions.
