Seven people enter a lift. The lift stops at three (unspecified) floor. At each of the three. no one enters the lift, but at leat one person leaves the lift. After the three floor stops, the lift is empty. In how many ways can this happen?
Why did my attempts fail?
let $x_{i's}$ be the number of persons leaving lift at once (any floor). Then problem becomes finding integer solution of: $$x_{1}+x_{2}+x_{3}=7\ \text{where,}\ \ 1\le x_{i}\le7$$
$\implies x'_{1}+x'_{2}+x'_{3}=4\ \text{where}\ \ x_i=1+x'_{i}$
The number of solutions: $\displaystyle{{4+3-1}\choose{4}}=15$, but this didn't work, please explain where I went wrong and give your solutions too.
My read is that your answer is correct provided that the seven passengers are indistinguishable.
If you can distinguish the passenger an inclusion-exclusion argument yields $3^7-3\cdot 2^7 + 3\cdot 1^7 =1806$ possibilities.