Quote by AdrianZ
One more question, Is it always possible to solve an equation like ax^{n}=b in S_{n}? When it's possible?

Quote by AdrianZ
Would you explain more please?

Explained with orders:
In S4, x^4 is either identity or a 3cycle (with order 3).
If a and b differ in order, but not by 3, there is no solution.
Explained with even and odd permutions:
(Do you know what even and odd permutations are?)
In S4, x^4 is always an even permutation.
If a is odd and b is even, then there is no solution.
Quote by AdrianZ
We found out that there are 1 onecycle, 6 different 2cycles, 8 different 3cycles and 6 different 4cycles in S_{4}. but if we add 1+6+8+6 it'd be equal to 21, not 24. How so?

Yes, you're missing 3 of them.
Did you already have them in your original solution?
As a challenge (when you find them), how should you count how many there are?