I thought I understood the fundamental counting principle, but I'm still hung up on why it does not count the permutations of the events as well. My misunderstanding was revealed when I tried to work out a textbook example before looking at how the example was worked out (trying to be an active reader by racking my brain before seeing the full solution). The textbook section containing the example is called "Distinguishable Permutations and Combinations".
Fundamental Counting Principle: Let $E_{1}, E_{2}, \ldots, E_{k}$ be a sequence of $k$ events. If, for each $i$, the event $E_{i}$ can occur in $m_{i}$ ways, then the total number of ways all the events may take place is the product $m_{1} m_{2} \cdots m_{k}$.
Textbook example: A little league baseball squad has six outfielders, seven infielders, five pitchers, and two catchers. Each outfielder can play any of the three outfield positions, and each infielder can play any of the four infield positions. In how many ways can a team of nine players be chosen?
My organization of the problem: We have 20 people on the squad. We want to determine how many 9-person teams can be chosen with one pitcher, one catcher, three outfielders, and four infielders. My strategy is to create four events based on the positions and then count the possibilities.
Let $E_1$ be the determination of the catcher, so $m_1 = C(2,1) = 2$.
Let $E_2$ be the determination of the pitcher, so $m_2 = C(5,1) = 5$.
Let $E_3$ be the determination of the three outfielders, so $m_3 = C(6,3) = 20$.
Let $E_4$ be the determination of the four infielders, so $m_4 = C(7,4) = 35$.
The fundamental counting principle I thought said that the total number of ways these events can occur is $2\cdot5\cdot20\cdot35$, but I thought that counted all $4!$ sequences of the events like $E_1,E_2,E_3,E_4$, $E_2,E_1,E_3,E_4$, and so on. As such I thought then that I should divide my result of $2\cdot5\cdot20\cdot35$ by $4!$.
The textbook I then saw does no such thing, so I'm confused. It seemed to me that the fundamental counting principle is calculating the number of different permutations when what we really want is the number of combinations of the four events.
What am I missing? When I think of a "without replacement" situation I have to ask myself if order matters. If it does not, then I need to divide off as I did above.
Best Answer
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.