Note how for non-interacting particles the probability is symmetric in its arguments. This is not true in general, and the order in which the positions occupy the argument slots of P ( n ) matters. Given a set of positions, the way that the N particles can occupy those positions is N ! The probability that those positions ARE occupied is found by summing over all configurations in which a particle is at each of those locations. This can be done by taking every permutation, π π , in the symmetric group on N objects, S N , to write ∑ ∑ π π ∈ ∈ S N P ( N ) ( r π π ( 1 ) , … … , r π π ( N ) ) . For fewer positions, we integrate over extraneous arguments, and include a correction factor to prevent overcounting, ρ ρ ( n ) ( r 1 , … … , r n ) = 1 ( N − − n ) ! ( ∏ ∏ i = n + 1 N ∫ ∫ d 3 r i ) ∑ ∑ π π ∈ ∈ S N P ( N ) ( r π π ( 1 ) , … … , r π π ( N ) ) This quantity is called the n-particle density function. For indistinguishable particles, one could permute all the particle positions, ∀ ∀ i , r i → → r π π ( i ) , without changing the probability of an elementary configuration, P ( r π π ( 1 ) , … … , r π π ( N ) ) = P ( r 1 , … … , r N ) , so that the n-particle density function reduces to ρ ρ ( n ) ( r 1 , … … , r n ) = N ! ( N − − n ) ! P ( n ) ( r 1 , … … , r n ) Integrating the n-particle density gives the permutation factor N P n , counting the number of ways one can sequentially pick particles to place at the n positions out of the total N particles. Now let's turn to how we interpret this functions for different values of n .