If all the people look alike to the elevator guy then it would be simply distributing 8 identical balls to 6 identical boxes and this would look like $x_1+x_2+x_3+x_4+x_5+x_6 =8$ and the solution would be $\binom{6+8-1}{8}=\binom{13}{8}=1287$
If the men and women are treated separately then the problem can be viewed as a combination of two simpler problems, i.e. distributing 5 red balls into 6 identical boxes and distributing 3 blue balls into 6 identical boxes. So, this would turn out to look like: $x_1+x_2+x_3+x_4+x_5+x_6 = 5$ corresponding to men being distributed to six floors and $y_1+y_2+y_3+y_4+y_5+y_6 = 3$ for the women. And the product of the number of solutions to the above equations gives us the final answer. And this would be $\binom{6+5-1}{5}*\binom{6+3-1}{3}=\binom{10}{5}*\binom{8}{3}=14112$