combination or permutation

Well, the only difference I see is in a combination problem we typically pick up more than 2 things from a group (so that their order does not matter)

Here the difference is we select only one teacher for each day. So when only one is selected permutation and combination would be the same since there is no ordering..

For the 1st day 6 choices, she picks up one - 6C1 = 6
2nd day, 5 choices - 5C1 = 5
3rd day 4 choices, 1 to be selected = 4C1 = 4

So, this essentially translated to 6! (which is also 6P6).

If the problem were to select 5 teachers from 6 in a single event, then we would have 6C5, but here we have 5 separate events and hence the difference.

The difference all lies in the exact wording of the question. The word "schedule" implies that order matters. For example, if you have to choose 2 classes out of five to take from school, choosing math and literature is the same as choosing literature and math. The difference is trivial, because we're just talking about WHICH classes you take. However, taking math on monday and literature on friday is different from taking literature on monday and math on friday. Once we add this second dimension of time, order matters. How many ways can we choose two classes out of five: combinations. How many ways can we schedule two classes out of five: permutations. It's all about the context.

The same thing is going on here. How many groups of 5 out of 6 teachers the principal can meet with is a separate question from how many ways she can schedule them. Imagine you're at your job and you know you have 5 clients to meet with that day. You go into work and ask your secretary if there have been any changes to the schedule. She says, "nope". But then, to your surprise, your 5pm client shows up at 8am. You go out there and say "What the hell? I thought you said there weren't any changes to the schedule" and she says "There weren't! it's still the same five clients today!" Something tells me you'd be looking for a new secretary.

Thanks Pete.

So the takeaway would be to pay close attention to the context and look for those keywords that hint at distinction between cases?