I am fixing $m,n,r,s$ such that $m + n = r + s$ from the beginning. I have the following $4$ conditions that should be satisfied : \begin{align*} r &= p + p'\\ s &= q + q'\\ m &= p + q\\ n &= p' + q' \end{align*} And I want to show that the following equation is satisfied under the previous four conditions: $${r + s \choose r + s - n} = \sum_{p,q,p',q'} {r \choose p} {s \choose q}. \quad \quad \quad (**)$$
How can I show this? Could someone help me please?
EDIT:
I feel like this is the equation I should prove but still I do not know how to do this:
$${r + s \choose m} = \sum_{p = 0}^m {r \choose p} {s \choose m - p}$$ but still I do not know how to prove this also.