# Probability that no two people get off elevator on same floor

Seven people enter the elevator on the first floor of a $12$ story building. What is the probability that no two will get off on the same floor? You may assume that all floors are equally likely and that no one gets off the elevator before it starts up.

I believe that answer is $\frac{11P7}{11^7}$, but the book lists the answer as $\frac{12P7}{12^7}$. Which is correct?

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This is funny. Isn't the difference between the two answers that you (reasonably) think none would get off on the first floor, while the book presumes it's equally probable as other possibilities? – anon Feb 6 '12 at 1:13
Note that "no one gets off the elevator before it starts up" does not preclude the possibility that some perverse person rides the elevator to a higher floor, then waits till it gets to floor 1 again and gets off. – David Mitra Feb 6 '12 at 1:19
"... that no one gets off the elevator before it starts up" should indicate that nobody gets off on the first floor, so the book is wrong. Or maybe they actually entered on the "ground floor", above which there are 12 numbered floors. Note that it's also necessary to assume that different people's destinations are independent. – Robert Israel Feb 6 '12 at 1:27
@DavidMitra: thanks, I incorrectly assumed that everybody had to take the path of least resistance to their destination – John Feb 6 '12 at 1:36

## 1 Answer

The problem statement specifies that there are 12 stories, so there are 12 other floors to go. Therein lies the misunderstanding...

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But wouldn't you say that the first floor is the first storey? – Gerry Myerson Feb 12 '12 at 5:35
Ah, that's a different story... There are different conventions in different countries. – Robert Israel Feb 12 '12 at 7:33
The street level floor is not considered to be the floor of the first story, in my view. It is the floor above it. – Jean-Victor Côté Feb 20 '12 at 2:47