I encounter the following problem: Let $q\geq t\geq 1$ and $m\geq 1$. Prove that $$E(q,t,m,n)=\sum_{j=0}^t (-1)^j \binom{q-t+j}{j} \binom{n}{m+t-j} \geq 0 \text{ for all }n\geq mq+m+q.$$ It is easy to see that $E(q,1,m,n)\geq 0$ if and only if $n\geq mq+m+q$, hence the later condition is necessary.
Also, it suffices to prove the inequality for $n=mq+m+q$.
I was able to show the inequality in some easy cases like $t\leq 4$, $q=t$ and $q\leq m+1$.
Also, the inequality is known to be true for $m=1$. But I have no clue how to tackle it in the general case.