How many movements are needed to reach all arrangements of a Rubik’s Cube?
According to my Google searches, there are $43,252,003,274,489,856,000$ possible arrangements, and the maximum number of moves required to solve is $20$.
Is that right, and how was $20$ calculated?
The number of legal positions is indeed $$ \frac{12! \cdot 8!}2 \cdot 2^{11} \cdot 3^{7} = 43\,252\,003\,274\,489\,856\,000 $$ which is derived in most mathematically-minded introductions to the cube.
The fact that each of these positions can be solved in $20$ moves (where turning a side 180° counts as one move) was discovered only in 2010 after an exhaustive computer search for positions that would need more. This used a combination of raw computer power (donated by Google, equivalent to one CPU running for 35 years) and clever tricks to speed up the search. There are details on http://cube20.org/
(If you only count 90° turns as moves, there are positions that require 26 moves to solve, found by similar methods 4 years later).