Summation without end value? $$\sum_{a+n-b=c}\binom{n}{a}\binom{n}{b}$$
What does this mean ? I found it here
It is meant to sum up from where to where ? 
 A: Assuming that $a,b, c$ are non-negative integers,
$$\begin{align}
\sum_{a+n-b=c}\binom{n}{a}\binom{n}{b}&=\sum_{a+n-b=c}\binom na\binom n{n-b}\\
&=\sum_{a^*}\binom na\binom n{c-a}\\
&=\binom {2n}c\qquad\qquad\text{by the Vandermonde Identity}
\end{align}$$
where $a^*$ indicates the following ranges for $a$:
$$\begin{cases} \begin{align}
0<c<=n: &\qquad a=0\quad\text{to}\quad
c\\    
n<c\leq 2n: &\qquad a=c-n\quad\text{to}\quad n
\end{align}\end{cases}$$
such that $0\leq a,\;c-a \leq n$.
A: In context , both $a$ and $b$ are whole numbers less than or equal to $n$ 
$$\sum_{a+n-b=c}\binom{n}{a}\binom{n}{b}=\sum_{a=a_{min}}^{a_{max}} \binom n a \binom n{(n+a)-c}$$
where $a_{max}=\min\{ n, c-n \}$ and $a_{min}=\max\{ 0, c-n \}$
A: You could write the sum over $a$ as  $$\sum_{a= \max(0, c-n)}^{\min(n,c)}\binom{n}{a}\binom{n}{a+n-c}$$ 
A: You sum it up on all possible values where (a + n - b=c) [In the context, n is a constant and c is a constant between 0 and 2n].  The conditions of a summation need not be a sequenced list.  You can do $\sum_{p|36; p >0} p$ to mean $1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36$.  This is in basically easier than writing $\sum_{i = 1}^{36} a_i$ where $a_i = i$ if $i|36$ and $a_i = 0$ if $i$ doesn't divide $36$.
So this is basically saying $\sum_{c = 0}^{2n}(\sum_{a = 0}^n \sum_{b = 0}^n{n \choose a}{n \choose b})$ *BUT ONLY IF b = n - c + a * or $\sum_{c=0}^{2n}(\sum_{a = \max(0, n-c)}^{\min(n,2n -c)} {n \choose a}{n \choose {n - c + a}})$ 
For example: if c = 0, this is a=0, b=n only.  If c = v < n, this is a= 0...v; b=n,(n-1)....(n -v).  If c = v > n, this is a = v-n...n; b = 0...(2n -v).   
It's just easier to write as $\sum_{c=0}^{2n}(\sum_{a + n - b = c} {n \choose a}{n \choose b})$ and once you get used to it it is easier to understand.
Basically $\sum_{c=0}^{2n}(\sum_{a + n - b = c} {n \choose a}{n \choose b})$ = $\sum_{a=0}^0{n \choose a}{n \choose {a + n}} + \sum_{a=0}^1{n \choose a}{n \choose {a + n -1}} + \sum_{a=0}^2{n \choose a}{n \choose {a + n -2}} + .... \sum_{a=0}^n{n \choose a}{n \choose {a}} + \sum_{a=1}^n{n \choose a}{n \choose {a - 1}} + .... + \sum_{a=n-1}^n{n \choose a}{n \choose {a -n + 1}} +\sum_{a=n}^n{n \choose a}{n \choose {a -n }}$
