So here's the first question:
If 7 people board an airport shuttle with only 3 available seats, how many different seating arrangement are possible? (Assume that 3 of the 7 will actually take the seats.)
So I think the answer is:
7! / 4! = 210
The 7! is equal to the total number of combinations right? The 4! represents the combinations of seated people.
So, the reason why I think the answer is 7! / 4! is because we need to uncount the different arrangements of the seated people because the seated people are all equivalent and indistinguishable i.e., it doesn't matter how they are seated. But why aren't we subtracting? Why are we dividing?
Second question:
If 3 of 7 standby passengers are selected for a flight, how many different combinations of standby passengers can be selected?
So the answer do this I think is:
7! / (4! * 3!)
So unlike the first question, the standby passengers and the flying passengers are both indistinguishable or rather order doesn't matter. But again, why aren't we subtracting? Why are we dividing?
Your answers are correct, but it appears that they are based on memorized formulas. A logical approach changes "I think?" to "obviously". First to fill the first seat there are 7 choices, to fill the second 6 choices and for the third 5 choices; altogether 7*6*5 [which is the same as 7!/4!]. Second to count the number of sets just divide the number of orderings 7!/4! by the number of ways to order a set of 3 objects 3!; obviously you have (7!/4!)/3! which is your answer.