Definition of Multiplication principle in Principles and Techniques in combinatorics by Chuan-Chong, Khee-Meng is given as:
Let $$\prod_{i=1}^rA_i=A_1\times\dots\times A_r=\{(a_1,\dots,a_r) | a_i\in A_i, i=1,\dots,r\}$$ denote the cartesian product of the finite sets $A_1,\dots,A_r.$ Then $$\left\vert \prod_{i=1}^r A_i \right\vert=|A_1|\times\dots\times|A_r|.$$
I want to know how this principle is applies in defining $P_r^n=n(n-1)\cdots2\cdot1$, ie, what are sets $A_1,\dots,A_r$ here?
Suppose you have a procedure that can be done in $n$ steps and that
Step 1 can be done $s_1$ ways Step 2 can be done $s_2$ ways no matter how the previous steps are done
....
Step n can be done $s_n$ ways no matter how the previous steps are done
Then the procedure can be done $$\prod_{k=1}^n s_k$$ ways.