Counting a one to one function: Where does the + 1 come from in n-m + 1 I am currently reading up on counting one-to-one function and understood that when I want to count a set with m elements to a set with n elements, I can use the product rule to count the number of combinations such that
n*(n-1)*(n-2)*(n-3)*.......(n - m + 1)

I can see why there would be a n - m, but I can not seem to reason where the + 1 might come from. Any help would be beneficial, thanks!
 A: Say I have 3 objects and I enumerate them 1,2,3, then the number of objects I have is (3-1)+1. So in your case you have n-(n-m+1)+1=m objects.
A: Let set $A$ has $m$ elements and set $B$ has $n$ elements.
Let denote the elements of $A$ as $a_0, a_1, \ldots , a_{m-1}$.
Let's assign values in the order from $a_0$ to $a_{m-1}$.
$a_0$ has $n-0$ choices.
$a_1$ has $n-1$ choices. (less one choices)
$a_2$ has $n-2$ choices.
We can see that $a_i$ has $n-i$ choices.
$a_{m-1}$ has $n-(m-1)$ choices.
A: You need m factors. If you rewrite your product into
$(n-0)\cdot(n-1)\cdot(n-2)\cdot(n-3)...(n - (m - 1))$
You see that it is exactly what you need. The point is that you start form $0$, not from $1$
A: It is because say for example n = 5 then 
(5 - 1) = 4 and so 4 + 1 = 5 
(5-2) = 3 and so 3 + 1 = 4 
(5-3) = 2 and so 2 + 1 = 3 
(5-4) = 1 and so 1+ 1 = 2 
(5-5) = 0 and so 0 + 1 = 1 
Therefore you need the plus one otherwise if you just had (n-m) the answer would always be one less than it is supposed to be and this is finding the factorial of n.
