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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?

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  • $\begingroup$ It is unclear. What do the arrangements mean? $\endgroup$ – QuIcKmAtHs Jan 21 '18 at 0:32
  • $\begingroup$ @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. $\endgroup$ – user13 Jan 21 '18 at 0:36
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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.

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  • $\begingroup$ 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 ? $\endgroup$ – user13 Jan 21 '18 at 15:01
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    $\begingroup$ @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. $\endgroup$ – Bram28 Jan 21 '18 at 15:20

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