Number of draws that contain at least $k$ red coins

Question

Assume there are $n$ coins in an urn from which $r$ are read. What is the number of draws of $r$ coins that contain at least $k$ red coins? It is obvious that there are $$\binom{r}{k}\binom{n-r}{r-k}$$ draws that contain exactly $k$ red coins. From which it is also clear that there are $$\binom{n}{r}-\binom{n-r}{r}$$ draws with at least $1$ red coin. To put it in formulas, I am interested in $$\sum_{j=k}^r\binom{r}{j}\binom{n-r}{r-j}$$ or at least a good lower bound on it.