We will assume that the 6 passengers are indistinguishable.  They aren't, but
the probability is the same whether they are considered indistinguishable or
not, and it's easier to consider them indistinguishable.

We will also assume that all 6 passengers get on the elevator at the same time
in the basement (parking level, not considered a floor) and the 8 floors are
numbered 1 (street level), 2, 3, 4, 5, 6, 7, and 8. 

A successful elevator trip to the top (8th) floor is one in which no two
passengers get off on the same floor. So we have 8 floors to pick 6 from for
each single passenger to get off at. That's '8 choose 6' or C(8,6) = 28 ways.

For the number of ways any number of passengers can get off at any floor, we
make a string of 6 stars to represent the 6 passengers: ******

Then we place 7 bars (1 less than the number of floors) among them, say, like
this random case:

 |*|**||*|*||*

This particular random arrangement of stars and bars represents the case where:

(a) The number of stars to the left of the 1st bar represents the number of
passengers who get off on the 1st floor. In this case, there are no stars to the
left of the first bar, so in this case, 0 passengers get off on the 1st floor.

(b) The number of stars between the 1st and 2nd bars represents the number of
passengers who get off on the 2nd floor. In this case, there is only 1 star
between them, so in this case, only 1 passenger gets off on the 2nd floor.

(c) The number of stars between the 2nd and 3rd bars represents the number of
passengers who get off on the 3rd floor. In this case, there are 2 stars between
them, so in this case, 2 passengers get off on the 3rd floor.

(d) The number of stars between the 3rd and 4th bars represents the number of
passengers who get off on the 4th floor. In this case, there are no stars
between them, so in this case, 0 passengers get off on the 4th floor.

(e) The number of stars between the 4th and 5th bars represents the number of
passengers who get off on the 5th floor. In this case, there is only 1 star
 between them, so in this case, only 1 passenger gets off on the 5th floor.

(f) The number of stars between the 5th and 6th bars represents the number of
passengers who get off on the 6th floor. In this case, there is only 1 star
between them, so in this case, 1 passenger gets off on the 6th floor.

(g) The number of stars between the 6th and 7th bars represents the number of
passengers who get off on the 7th floor. In this case, there is only 1 star
between them, so in this case, only 1 passenger gets off on the 7th floor.

(h) The number of stars to the right of the 7th bar represents the number of
passengers who get off on the 8th floor. In this case, there is only 1 star to
the right of the 7th bar, so in this case, 1 passenger (the last one), gets off
on the 8th floor.

******||||||| is the case when all 6 passengers get off on the 1st floor.

**|**|**||||| is the case where 2 get off on each of the first 3 floors and no
passenger goes any higher. 

|||||||****** is the case where all 6 passengers get off on the 8th floor.

|||***|***||| is the case where 3 get off on the 4th floor and the other 3 get
off on the 5th floor.

For every case, there are 6+7=13 things in a row, counting bars and stars.
There are 6 indistinguishable stars and 7 indistinguishable bars, so the
number of cases of passengers getting off is 13!/(6!*7!) = 1716.

Answer: 28 successful elevator trips out of 1716 possibilities, or 28/1716 =
7/429. 

Edwin