For the well-known exponential complexity, computing the correlation function for general many-body wavefunctions is a major challenge. We study the ground state $n$ order correlation function of Tonks-Girardeau (TG) gas. Based on the free fermion wavefunction and the Bose-Fermi mapping method, we obtain the exact ground-state wavefunction of TG gas. Using the properties of the Vandermonde determinant and the Toeplitz matrix, the $n$th order correlation function is formulated as the $(Nn)$th order Toeplitz determinant whose elements depend on the 2$(Nn)$ sign functions. , which is computed analytically.By reducing the integral over the domain $[0,2\pi]Converting $ to a sum of integrals over several independent domains finally gives the explicit form of the Toeplitz matrix elements. As an application, we derive a simple formula for the reduced two-body density matrix and discuss its properties. The corresponding natural orbitals and their occupancy distributions are plotted. Furthermore, we give a concise formula for the reduced three-body density matrix and discuss its properties. A second measurement in succession shows that atoms appear in regions where atoms are most likely to be in the first measurement.
