for positive integers $m$, $n$ such that $m \le n$ $${m+n \choose m} = {m \choose 0}*{n \choose 0} + {m \choose 1}*{n \choose 1} +......+ {m \choose m}*{n \choose m}$$
I was trying out some random examples and this seemed true for them. Is this equation true for all values of m and n? If yes, can it be proven?
As ${m\choose k}={m\choose m-k}$ you can write the conjectured identity as $${m+n\choose m}=\sum_{k=0}^m{m\choose m-k}{n\choose k}\ .$$ This can be proven as follows: You can choose a team of $m$ people from $m$ boys and $n$ girls by choosing first the number $k\in[0\,..\,m]$ of girls in the team and then choose the boys in ${m\choose m-k}$ ways and the girls in ${n\choose k}$ ways.
