if I understand correctly:

2 elements: 2 permutations

a,b

b,a

3 elements: 6 permutations

a, b, c

a, c, b

b, a, c

b, c, a

c, a, b

c, b, a

You can see that each element takes a turn out front, then the remaining 2 elements go through their 2 permutations.

So it follows that if we add a fourth element, then each will take a turn out front, and the remaining three will go through their 6 permutations.

So it is basically

f(n) = n*f(n-1)

where f(1)=1