TITLE: what's the summation of this finite sequence?
QUESTION [6 upvotes]: $a$ and $b$ are positive integers.
The summation is 
$$\sum\limits_{x = 1}^a {x\left( {\begin{array}{*{20}{c}}
{a + b - x}\\
b
\end{array}} \right)} .$$
Any closed-form expression?
I thought it should have. And maybe there is some physical meaning behind it.

Sorry, I've simplified the problem, and now it becomes easier.

REPLY [6 votes]: This is the binomial identity $\sum_{m=0}^n\binom{m}{j}\binom{n-m}{k-j} = \binom{n+1}{k+1}$ with $j = 1$, $n = a+ b$ and $k = b+1$.