Question
Let's say we have $n$ disinct types of objects. For each category $i$ we have $n_{i}$ of them where $i \in \{1,2, \dots n\}$ and $n_{1}+n_{2}+ \dots n_{n} = m$ . In how many ways can a set of $r$ objects be chosen from the above $m$ objects?
Related Question
I always see another version of this problem where for each category we have infinitely many objects and the number of ways of choosing $r$ objects form them reduces to finding number of non negative integer solution of the following:
$n_{1} + n_{2} + \ldots + n_{n} = r$
and that will be ${r+n-1 \choose n-1 }$
