fundamental counting principle, permutations, counting arrangements of events

Solution 1:

The fundamental counting principle does not count the number of different permutations. What it counts is the size of a cartesian product of finitely many finite sets:

$$E_1\times E_2\times\cdots\times E_k$$

Unfortunately, the textbook uses a somewhat misleading terminology, and talks about a "sequence of $k$ events", which may -- and clearly does -- cause some confusion as to the role of the order of the "events". However, this fundamental principle of counting does not count ordered $k$-tuples; it counts sets of ordered $k$-tuples; all it cares about is in how many ways you can choose each member of a $k$-tuple. Once you have them chosen, their order does not matter anymore, just as the order by which a set of players of a baseball team enter the dressing room at the end of the match, does not really matter.