For the sum in the binomial theorem, what if the upper limit was different than n in the rest of the summand?

Question

For the sum in the binomial theorem, what if the upper limit was different than n in the rest of the summand? Like this: $$ \sum_{k=0}^{m}\binom{n}{k}x^{k}y^{n-k} $$ Does it have a closed form at all? Or even anything like where x or y equals 1 or something? Or is this the same as trying to find $\sum_{0}^{m}\binom{n}{k}$, where it's impossible?