I want to express how many ways you can arrange $k$ things in $n$ places.
$$\prod\limits_{k=k}^n k = k (k+1) (k+2)\cdots(n-1) n$$
Edit (added) { The example from which I started thinking about this was placing 3 different letters(always placing each one, and with no repetition, so after arranging,two spaces are left empty) in 5 places. The equivalent form of what I'm asking for would be $\frac{n!}{(n-k)!}$ I think. }
1) Is this a valid product notation (mainly referring to the $k = k$)? If not, why?
2) Is the expression correct for that purpose?
If not: a) how it should be? b) Does it express other thing?
If you have $k$ distinguishable things and $n$ distinguishable places to put them you have $n$ choices to place the first thing. When this first choice has been made you have $n-1$ choices left to place the second thing, and so on. When it comes to place the $k$'th thing there are $n-(k-1)$ empty places left. All in all there are $$N=n(n-1)(n-2)\ldots\bigl(n-(k-1)\bigr)$$ possibilities. Now it comes to write this number $N$ without using "$\ldots$". Since the letters $n$ and $k$ already have been used as parameters of the problem we have to introduce a new symbol, say: $j$, as "multiplication index". In this way we arrive at $$N=\prod_{j=0}^{k-1} (n-j)\ .$$ Another way of representing this number $N$ would be $$N={n!\over(n-k)!}\ .$$