In a book I had read
The number of ways in which n different things can be arranged in r different groups is $n!\dbinom{n-1}{r-1}$
Question 1.
So I had interpreted it like this
First n things are arranged by $n!$ ways then then the gaps are chosen for separators(that separates different groups).Number of gaps is $n-1$ assuming no group is empty and we have to choose $r-1$ out of them, so $n!\dbinom{n-1}{r-1}$.
So have I interpreted it correctly?
If yes , then if groups can be empty so number of ways will be $n!\dbinom{n+1}{r-1}$ as in this case number of gaps will be $n+1$.
Question 2
Why cant I interpret it like this we have $n+r-1$ things in which $r-1$ are alike so the number of ways will be $\frac {(n+r-1)!}{(r-1)!}$(which is not correct according to the book).