separation of semi-distinguishable objects.

Question

I have $n_1$ objects of type $1$, $n_2$ objects of type $2$, ..., $n_k$ objects of type $k$.

Now, What are the numbers of ways of making $p$ objects out of these $n=\sum n_i$ semi-distinguishable objects where $$\left\lfloor\frac{n}2\right\rfloor+1\le p\le n\;.$$