We'll treat the problem only for a two-particle system.
Consider two identical particles which can be in the individual states
,which include the spin-dependence and are orthogonal.
The physical state of the system is:

Suppose we want to measure on each particle the physical quantity
![{\displaystyle G\left|g_i\right\rangle=g_i\left|g_i\right\rangle \quad \left[G,P_{21}\right]=0 }](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/56e2385da87d935ffc7561e2ee090a14a540c7c8)
What is the probability of finding

for one particle and

for the other?
The final state is:

The desired probability is given by

for bosons and

for fermions, that is:

The term

is called
direct term, while the term

is called
exchange term:

This shows the appearance of a sort of interference phenomenon when we're dealing with identical particles.