Answer
There are C(n + r − q1 − q2 ...− $q_{r}$ −1, n − q1 − q2 −...− $q_{r}$ ) selections
Work Step by Step
The order of selection does not matter. Thus we use $C$().
There are n objects of r different types.
There are at least $q_{1}$ objects of type one, at least $q_{2}$ of type 2,.. at least $q_{r}$ of type r. We select $q_{1}$ objects of type one, $q_{2}$ of type 2,.. $q_{r}$ of type r.
So there are, n-$q_{1}-q_{2}...-q_{r}$ that has to be selected from r different types.
Using the definition of combination with repetition allowed, we obtain that there are
C(n + r − q1 − q2 ...− $q_{r}$ −1, n − q1 − q2 −...− $q_{r}$ ) selections
