An elevator starts moving from the ground floor with $8$ passengers, so that all passengers get off the elevator until the sixth floor An elevator starts moving from the ground floor with $8$ passengers, so that all passengers get off the elevator until the sixth floor.
Assuming passengers are the same, then

*

*In how many ways  is it possible for the passengers to get off the elevator?


*If $3$ passengers are women and the rest are men, then in how many ways  is it possible for the passengers to get off the elevator?

I think the answer to the first question is equivalent to the number of $6$-tuples of non-negative integers whose sum is $8$,
$$\sum_{1\le i\le6}^{ }x_{i}=8 \implies \text{the number of such solutions}=\binom{8+6-1}{6-1}=1287$$
The answer to the second question is equal to the number of solutions to the following system of equations:
$$\sum_{i=1}^{6}x_{i}=3\;\;\;,\;\;\;\sum_{i=1}^{6}x_{i}=5$$
Where $x_i$'s are non-negative integers, so the answer is $$\binom{3+6-1}{6-1}\binom{5+6-1}{6-1}=\frac{8!}{3!5!}\frac{\left(10\right)!}{5!5!}=14112$$
I want to check the validity of my answers.
 A: What you have looks good mathematically. My only concern would be the wording of the question. In particular, is the sixth floor six floors higher than the ground floor (British English) or only five (American English)? Well, to tell you the truth, in all this excitement I kind of lost track myself.
The point is that if everyone gets on on the ground floor, no-one gets off on that floor, so if the ground floor and first floor are the same, you should be looking at $5$-tuples, not $6$-tuples. The use of the word "elevator" rather than "lift" suggests that is the case I think.
A: As far as i have understood from your question , the passanges must get off either first , or second or third or fourth or fifth floor because when they arrive to sixth floor there must not be any passengers and no passenger get off in ground floor.
Hence we have $5$ different floor ,i.e, $5$ different boxes.
$\color{red}{KNOWLEDGE=}$ Selecting objects is the same as placing object
Case I-) It is said that passengers are indistinguishable , so by star and bars $$C(8+5-1,4)=495$$
Case II-) Lets make use of generating fuctions for fancy .
$$C(5+3-1,4) \times C(5+5-1,4)=4410$$
