Problem Solving

Hi there! I'm getting really confused on when to divide a factorial by TWO factorials, versus when to only divide by ONE factorial. I've been following the MGMAT books and understand the Anagram analogy, but when I apply the principles to "real life" (aka, practice problems), my understanding doesn't always match up with what the answers actually are. I thought that we only had divide when there was a set of "repeats" in the factorial. Case in point: two questions below. Why do I divide one truffle box-making example by another factorial, but do NOT do this with the passengers of the plane?

1. Truffles:
"Amy and Adam are making boxes of truffles. They have an unlimited supply of 5 types. Each box holds 2 truffles of different types. How many boxes can they make"?

I thought the answer was 5!/3! (I chose 3 because there are 3 "no's", or truffles that don't make it into the box).

BUT, the answer in the book is: 5!/(2!*3!) = 10. I understand that this is because it's "2 yesses" (2 truffles made it into the box) and "3 no's" (3 truffles did not) but since the truffles are of different types, couldn't I technically categorize this as "1-2-NNN" (N = no) meaning it would just be 5!/3! ? My reasoning for that is supported (at least, to me) by the following plane passenger question:

2. Plane Passengers:
"If 7 people board an airplane with only 3 available seats, how many different seating arrangements are possible?"
The answer here is 7!/4! (aka, 1-2-3-NNNN) (N = no seat for person). Why is it this, and NOT "7!/(3!*4!)

....

I hope that makes sense - would definitely appreciate any explanations, rules to follow, examples to keep in mind, etc! Thank you!