**Question**: How to calculate this summation $S=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k} $? Where $m<n_1,m<n_2$

**Remark1**: When $a=b$, I know the above summation $S=a^m\sum_{k=0}^m {n_1\choose k} {n_2\choose m-k} =a^m {n_1+n_2\choose m} $.

**Remark2**: This summation looks somewhat similar to the usual binomial formula $\sum_{k=0}^m a^k b^{m-k} {m\choose k} =(a+b)^m$. So is there also a similar formula for $S$?