Number of ways getting out of the elevator? At the ground floor of a building 6 men and 5 women get into the elevator. The elevator gets up and stops at each floor and stops at the floor 8. In how many ways can the people get out if all men are considered identical and all women are considered identical and it is possible that at one floor no one gets out. We have 11 people. So are there $8^{11}$ ways?

Hint. Let $m_i$ and $w_i$ be respectively the number of men and the number of women that get out at floor $i$. Then the desired number of ways is the product of the numbers of the non-negative integer solutions of these two equations: $$\sum_{i=1}^8 m_i=6\quad\mbox{and}\quad\sum_{i=1}^8 w_i=5.$$ See Stars and Bars#Theorem_two) technique.