Number of ways to choose same number of elements from two different sets
Sometimes a summation of products of binomial coefficients can be simplified using  Vandermonde's identity. 

In your case, Assuming $p\ge q$ we have $\sum^q_{k=0}{p\choose k}{q\choose k}=\sum^q_{k=0}{p\choose k}{q\choose q−k}$, which by Vandermonde equals $p+q\choose q$.

Actually this equality can be understood with elementary combinatorial arguments. Put the $p+q$ elements in one box. Now grabbing $q$ elements from the box means you have selected some $i$ elements from $S$ and $q−i$ elements from $R$. Or unselected $i$ elements from $R$ which is the same.