SOLUTION: Seven passengers together enter an elevator on the first of an 11-floor building. Assume that the egress pattern to floors 2, 3, · · ·, 11 is the same. What is the probability tha

We make two choices, one that does not involve order and one that does.

1.  The first is to choose 7 floors for them to be getting off on.

Nobody is going to get off at the first floor because they wouldn't 
have gotten on, so there are 10 floors they could be getting off at.

Order does not matter here because the order in which we pick the 7
floors for them to get off at does not matter.  For instance if we 
pick floors 2,4,6,7,8,9 and 10, it doesn't matter whether we pick 
floor 4 first, floor 7 second, floor 8 third, etc., ...  or whether 
we pick floor 9 first, floor 2 second, etc., it's the same 7 floors.

So there are 10C7 ways to choose the 7 floors.

Now for each choice of 7 floors, we must arrange the 7 people to get 
off at those floors.  For every combination of 7 floors, we can 
arrange the people in 7P7 or 7! ways.  This involves order because 
the case when person A gets off on floor 9 and person B gets off on 
floor 4, that is different from the case when person A gets off on 
floor 4 and person B gets off on floor 9.     

Answer (10C7)(7P7) = 120*5040 = 604,800.

Edwin