How to think about a basic combinatorial question Let's say we have 20 individuals and need to assign them each one of 20 jobs. This is a pretty standard introductory level question in the combinatorics background presented in a probability theory course.
One solution says: line up the individuals in a row, then for the first one, you have 20 choices of job, for the second you have 19, and so on. Meaning there are 20! potential ways to assign the jobs.
What I find confusing about this example is the fact that you actually started with 20! ways to order the individuals in a row to begin with. One might be tempted to think that, due to the multiplication rule, there are actually $(20!)^2$ ways to assign the jobs.
Further reflection reveals: yes, indeed, there are $(20!)^2$ person/job pairs where we care which appears first, second, third, etc in the row. But since we're not being asked to designate a first, second, third, etc. pair we need to correct for the over-counting. Each set of 20 person-job pairs can be ordered $20!$ ways, so we have the correct answer as: $(20!)^2 /  20! = 20!$
Is the above reasoning sound? When we approach a problem like this one by imagining lining up the people or objects to be labeled/chosen, are we "automatically"/implicitly adjusting for the over-counting that I've somewhat painfully accounted for explicitly above?
 A: Yes, all of your reasoning looks sound, these are good things to be thinking about.

When we approach a problem like this one by imagining lining up the people or objects to be labeled/chosen, are we "automatically"/implicitly adjusting for the double-counting that I've somewhat painfully accounted for explicitly above?

Yeah, that's a good way to put it. More generally, whenever we order $n$ objects without loss of generality, we are really multiplying by the $n!$ ways to order them, and then dividing by $n!$ because their order doesn't matter, so the net effect $n! / n! = 1$ cancels out, exactly as you say.
A: Yes, it is natural to follow the approach of "imagining lining up the people or objects to be labeled/chosen " and then "adjusting for the over-counting".  For brevity, call this approach "lining-up and dividing".
Of course, it is also natural to "just choose an order and then count", since "the count doesn't depend on the order", as Qiaochu Yuan remarked.
Which way is more "natural" is up to debate. It might not be obvious at all a priori which way is more natural.

We can understand "lining up and dividing" more by checking other situations.
A classical example is how we can find the number of combinations of $k$ objects out of $n$ objects, ${n\choose k}$. What we usually do is to line up each combination of $k$ objects, thus counting $n\times (n-1)\times\cdots\times (n-k+1)$ cases. Then we realize that the same combination can be lined up in $k!$ ways. We conclude that ${n\choose k}$ is $n(n-1)\cdots(n-k+1)/k!$.
It is reasonable to see the counting approach right above is also "lining up and dividing". We can even argue this approach is the natural approach to compute ${n \choose k}$.
There are many similar situations where it is difficult to count without duplicates. Lining up first provide an initial picture that is often clear, precise and easy to manipulate.
Personally, I go back to this "lined up" state to put myself back on correct and firm starting point when I got confused. I do get confused from time to time even though I have done hundreds or more counting problems. Also, this "lined up and dividing" approach is often easier to be understood when I communicate my solutions.

More generally, "counting by division" or "Rule of
division" is
a fundamental counting methods. It is often preceded by "lining up".
A: Thanks for everyone's input on this.
I think the key insight here is to be found in answering the question: what are we actually doing when we line things up in a row?
Let's take the first job and assign a person to it. That's one sub-experiment for which there are $20$ choices. Let's take the second job, or even the 8th job for that matter, and assign another individual. This can be done $(n-1)$ ways. The multiplication principle says that sub-experiments can be completed in any order (ultimately due to the commutativity of multiplication).
So when we line things up in a row, what we're really doing is encoding the sequential performance of sub-experiments one by one. There's no need to consider the ordering of the items in the row and then adjust for the double counting because we're free to perform the sub-experiments in any order.
There's one additional subtlety here that's worth mentioning. The labeling principle, as defined by James Tanton (see here).
To elaborate, the labeling principle entails creating a label for each of the $n$ objects in question. In cases where this seems unnatural (for example, in the canonical example of appointing a president, treasurer, and secretary among a club of 10 people), you simply label any residual objects as "not selected".
There's a subtle shift that now occurs. In counting problems generally, you're choosing from $n$ items. E.g., you have $n$ people to name president and you select one. With the labeling principle/approach, you turn that around. Now you take an individual object and choose from the $n$ available labels.
The question I posed to start this thread (20 people, 20 jobs) obfuscates this dynamic, which I think adds to the confusion.
