Question 1
Four positive integers $m,n,p,q$ with $p\lt m$ and $q\lt n$ are given. Consider $A(0,0)$, $B(p,0)$, $C(m,q)$, $D(m,n)$. Paths $f:A\to D$ and $g:B\to C$ consist of unit steps right or up. Let $S$ be the number of ordered pairs $(f,g)$ with no common points. Prove
$$S=\binom{m+n}{n}\binom{m+q-p}{q}-\binom{m+q}{q}\binom{m+n-p}{n}.$$
$$S=\binom{m+n}{n}\binom{m+q-p}{q}-\binom{m+q}{q}\binom{m+n-p}{n}.$$