Label the people $A, B,C$.
Person $A$ has 6 floors she can get off at.
Person $B$ has only 5 floors she can get off at (this is because person $B$ cannot get off on the same floor as person $A$).
Person $C$ then has $4$ floors she can get off at (since she can not get off at the same floor as either person $A$ or $B$).
Thus, there are $4\cdot 5\cdot 6=120$ ways these three people can get off at different floors. Now, assuming there were no constraints on which floors the people excited, each person would have a total of 6 floors to choose to get off at, for a combined total of $6^{3}=216$ ways the people could have chosen to get off.
Therefore, the desired probability is $\frac{120}{216}=\frac{5}{9}$.