I found this paragraph and I'm having trouble reading it, I understand that the total of all possible arrangments equal to A1*A2*A3...An but I don't understand how we get there from this:
If A1, A2...Ak are sets so that #Ai=ni>=1 for i=1...k, so cartesian product o A1 x A2 x ... x Ak, consisting of all vectors form (a1, a2...ak) with ai E Ai, has a total of n1*n2...nk elements, meaning:
(A1 x ... x Ak) = n1 * n2 ... nk
The paragraph explains the meaning.
The operator "$\#$" means "number of elements of" so if $\#(A)= 3$ (say) and $\#(B)= 5$ then $$ \#(A \times B)= 3 \times 5 = 15. $$
To see why, suppose $A = \{a,b,c,d,e\}$ and $B = \{1,2,3\}$ and convince yourself that there are $15$ (letter,number) pairs.