<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> -body antibunching in a degenerate Fermi gas of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>He</mml:mi><mml:none /><mml:mo>*</mml:mo><mml:mprescripts /><mml:none /><mml:mn>

A key observable in investigations into quantum systems are the $n$-body correlation functions, which provide a powerful tool for experimentally determining coherence and directly probing the many-body wavefunction. While the (bosonic) correlations of photonic systems are well explored, the correlations present in matter-wave systems, particularly for fermionic atoms, are still an emerging field. In this work, we use the unique single-atom detection properties of $^3$He* atoms to perform simultaneous measurements of the $n$-body quantum correlations, up to the fifth-order, of a degenerate Fermi gas. In a direct demonstration of the Pauli exclusion principle, we observe clear anti-bunching at all orders and find good agreement with predicted correlation volumes. Our results pave the way for using correlation functions to probe some of the rich physics associated with fermionic systems, such as d-wave pairing in superconductors.