probability - distinguishable vs indistinguishable objects/persons I am solving one simple problem just for fun. 
The problem is as follows.  
$n$ people have to be seated randomly in a cinema hall which has $(n+k)$ seats.
What is the probability that in this process, some fixed $m$ places ($m \leq n$) get occupied/taken?  
The book gives this answer:
${{n} \choose {m}} \cdot {{n-k-m}\choose{n-m}} / {{n+k} \choose {n}}$  
There are 2 things I don't like here:  
1) I think that ${{n-k-m}\choose{n-m}}$ is obviously a typo and should read ${{n + k-m}\choose{n-m}}$ 
Why? Well to say the least $n-k-m \leq n-m$ so this binomial coefficient does not make much sense here. 
2) This answer seems to treat the $n$ people as indistinguishable which is strange. Since these are $n$ people/persons these are $n$ different people (we can never have identical people, let alone $n$ of them, even twins are different persons). Thus it makes more sense to treat the people as distinguishable. So the order in which the people take the places also matters, right?
But in that case I am getting this answer:  
${{n} \choose {m}} \cdot m! \cdot {{n + k-m}\choose{n-m}} \cdot (n-m)! / ( {{n+k} \choose {n}} \cdot n! ) $  
What is your opinion of these two items?
Am I correct in both or at least in one of them?  

The term $\binom{n}{m}$ counts the number of ways $m$ of the $n$ people can occupy the specified $m$ seats.  That leaves $n + k - m$ seats available.  The remaining people can occupy $n - m$ of them.  Hence, the number of favorable cases is 
$$\binom{n}{m}\binom{n + k - m}{n - m}$$
as you suspected.  The denominator $\binom{n + k}{n}$ represents the number of ways the $n$ people can select $n$ of the $n + k$ seats in the theater.
The people are distinguishable in the sense that we are selecting which $m$ of the $n$ people sit in the designated seats.  What does not matter is which particular person sits in which particular seat, just which seats are occupied.
