# The elevator starts with seven passengers and stops at ten floors

The elevator starts with seven passengers and stops at ten floors. The various arrangements of discharge may be denoted by symbols like $(3,2,2)$, to be interpreted as the event that three passengers leave together at a certain floor. two other passengers at another floor, and the last two at still another floor. Find the probabilities of the fifteen possible arrangements ranging from $(7)$ to $(1, 1, 1, 1, 1, 1, 1)$.

The answer, for example for $P(5,2)= \frac{10!} {8!*1!*1!} *\frac{7!} {5!*2!}*10^{-7}$

Please, can you explain the first term. I mean why $10$ floors are divided into groups with $8, 1$ and $1$ floors. Why not $9,1$ or anything else?

• It is unclear. What do the arrangements mean? – QuIcKmAtHs Jan 21 '18 at 0:32
• @XcoderX But... I wrote it. In this case it means that 5 people leave together in certain floor and another 2 together at the other floor. – user13 Jan 21 '18 at 0:36

It's because the $(5,2)$ discharge means that all passengers get off on one of two floors. So, that could be $5$ in floor $1$, and $2$ on floor $2$, or $5$ on floor $1$ and $2$ on floor $3$, or $5$ on floor $2$, and $2$ on floor $1$ ... etc.
There are $\frac{10!} {8!*1!*1!}$ ways to pick those two floors: $8$ floors where no one gets off, $1$ floor where $5$ people get off, and $1$ floor where $2$ people get off.
• Following, the logic for $(5, 1, 1)$ there must be $7$ floors where no one gets off, $1$ for $5$ people, $1$ for the second and $1$ for the third ? – user13 Jan 21 '18 at 15:01
• @SargisIskandaryan Yes, that's what $(5,1,1)$ means, but note that picking, say, floor $3$ for the 'second' floor, and floor $6$ for the 'third' is of course the same as picking $3$ for the 'third' and $6$ for the 'second', since the same number of people get off on those two floors. That is, those two floors are effectively indistinguishable, meaning that instead of using the term $\frac{10!}{7!1!1!1!}$ you should be using $\frac{10!}{7!1!2!}$. Think about it this way: There are $7$ floors where no one gets off, $1$ floor where $5$ people get off, and $2$ floors where $1$ person gets off. – Bram28 Jan 21 '18 at 15:20